diff options
Diffstat (limited to 'FSF-2020/linear-algebra/vector-spaces/Vector-Spaces')
55 files changed, 1987 insertions, 0 deletions
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Basis_of_a_dual_vector_space.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Basis_of_a_dual_vector_space.py new file mode 100644 index 0000000..630670e --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Basis_of_a_dual_vector_space.py @@ -0,0 +1,126 @@ +from manimlib.imports import *
+class DualVectorSpace(ZoomedScene):
+
+ def construct(self):
+ c1 = Ellipse(radius = 2,color=BLUE)
+ c2 = Ellipse(radius = 2,color=YELLOW)
+ c1.rotate(np.pi/2)
+ c2.rotate(np.pi/2)
+ c1.shift(2*LEFT+0.6*UP)
+ c2.shift(2*RIGHT+0.6*UP)
+ c1.scale(2)
+ c2.scale(2)
+ self.play(ShowCreation(c1))
+ self.play(ShowCreation(c2))
+ dot1 = SmallDot(color=BLUE).shift(2*LEFT+2*UP)
+ dot2 = SmallDot(color=BLUE).shift(2*LEFT+1.5*UP)
+ dot3 = SmallDot(color=BLUE).shift(2*LEFT+1*UP)
+ dot4 = SmallDot(color=BLUE).shift(2*LEFT+0.5*UP)
+ dot5 = SmallDot(color=BLUE).shift(2*LEFT)
+ dot6 = SmallDot(color=BLUE).shift(2*LEFT+0.5*DOWN)
+ dot7 = SmallDot(color=BLUE).shift(2*LEFT+1*DOWN)
+ text1 = TextMobject(r"$V$").scale(0.6).shift(3*UP+2*LEFT)
+ text2 = TextMobject(r"$V^* = \{T:V\rightarrow F\}$").scale(0.6).shift(3*UP+2.5*RIGHT)
+ self.play(ShowCreation(dot1),ShowCreation(dot2),ShowCreation(dot3),ShowCreation(dot4),ShowCreation(dot5),ShowCreation(dot6),ShowCreation(dot7))
+ v1 = TextMobject(r"$v_1$").scale(0.5).shift(2.2*LEFT+2*UP)
+ v2 = TextMobject(r"$v_2$").scale(0.5).shift(2.2*LEFT+1.5*UP)
+ v3 = TextMobject(r"$v_3$").scale(0.5).shift(2.2*LEFT+1*UP)
+ v4 = TextMobject(r"$v_4$").scale(0.5).shift(2.2*LEFT+0.5*UP)
+ v5 = TextMobject(r"$v_5$").scale(0.5).shift(2.2*LEFT)
+ v6 = TextMobject(r"$v_6$").scale(0.5).shift(2.2*LEFT+0.5*DOWN)
+ v7 = TextMobject(r"$v_7$").scale(0.5).shift(2.2*LEFT+1*DOWN)
+ self.play(ShowCreation(v1),ShowCreation(v2),ShowCreation(v3),ShowCreation(v4),ShowCreation(v5),ShowCreation(v6),ShowCreation(v7))
+ self.play(ShowCreation(text1))
+ dot9 = SmallDot(color=YELLOW).shift(2*RIGHT+2*UP)
+ dot10 = SmallDot(color=YELLOW).shift(2*RIGHT+1.5*UP)
+ dot11 = SmallDot(color=YELLOW).shift(2*RIGHT+1*UP)
+ dot12 = SmallDot(color=YELLOW).shift(2*RIGHT+0.5*UP)
+ dot13 = SmallDot(color=YELLOW).shift(2*RIGHT)
+ dot14 = SmallDot(color=YELLOW).shift(2*RIGHT+0.5*DOWN)
+ dot15 = SmallDot(color=YELLOW).shift(2*RIGHT+1*DOWN)
+ self.play(ShowCreation(dot9),ShowCreation(dot10),ShowCreation(dot11),ShowCreation(dot12),ShowCreation(dot13),ShowCreation(dot14),ShowCreation(dot15))
+ v9 = TextMobject(r"${T_1}$").scale(0.5).shift(2.2*RIGHT+2*UP)
+ v10 = TextMobject(r"${T_2}$").scale(0.5).shift(2.2*RIGHT+1.5*UP)
+ v11 = TextMobject(r"${T_3}$").scale(0.5).shift(2.2*RIGHT+1*UP)
+ v12 = TextMobject(r"${T_4}$").scale(0.5).shift(2.2*RIGHT+0.5*UP)
+ v13 = TextMobject(r"${T_5}$").scale(0.5).shift(2.2*RIGHT)
+ v14 = TextMobject(r"${T_6}$").scale(0.5).shift(2.2*RIGHT+0.5*DOWN)
+ v15 = TextMobject(r"${T_7}$").scale(0.5).shift(2.2*RIGHT+1*DOWN)
+ self.play(ShowCreation(v9),ShowCreation(v10),ShowCreation(v11),ShowCreation(v12),ShowCreation(v13),ShowCreation(v14),ShowCreation(v15))
+ self.play(ShowCreation(text2))
+ line1 = Line(start=dot1,end=dot9,stroke_width=0.95)
+ line2 = Line(start=dot2,end=dot10,stroke_width=0.95)
+ line3 = Line(start=dot3,end=dot11,stroke_width=0.95)
+ line4 = Line(start=dot4,end=dot12,stroke_width=0.95)
+ line5 = Line(start=dot5,end=dot13,stroke_width=0.95)
+ line6 = Line(start=dot6,end=dot14,stroke_width=0.95)
+ line7 = Line(start=dot7,end=dot15,stroke_width=0.95)
+ self.play(ShowCreation(line1),ShowCreation(line2),ShowCreation(line3),ShowCreation(line4),ShowCreation(line5),ShowCreation(line6),ShowCreation(line7))
+ self.wait(1.5)
+ rect1 = Rectangle(stroke_width=0.5,width=1,height=1.9).set_fill(color=BLUE,opacity=0.3)
+ vgroup1 = VGroup(dot3,dot4,v3,v4)
+ rect1.surround(vgroup1)
+ self.play(ShowCreation(rect1))
+ text3 = TextMobject(r"Basis of $V$ = $\{v_3, v_4\}$").shift(4.6*LEFT+1*UP).scale(0.5)
+ self.play(ShowCreation(text3))
+ self.wait(1.5)
+ rect2 = Rectangle(stroke_width=0.5,width=1,height=1.7).set_fill(color=YELLOW,opacity=0.3)
+ vgroup2 = VGroup(dot11,dot12,v11,v12)
+ rect2.surround(vgroup2)
+ self.play(ShowCreation(rect2))
+ text4 = TextMobject(r"Basis of $V^*$ = $\{{T_3},{T_4} \}$").shift(4.6*RIGHT+1*UP).scale(0.5)
+ self.play(ShowCreation(text4))
+ self.wait(2.5)
+ v9.move_to(3*LEFT+3*UP).scale(1.2).set_color(YELLOW)
+ colon = TextMobject(":").shift(3*UP+2.6*LEFT)
+ vgroup3 = VGroup(line1,line2,line3,line4,line5,line6,line7)
+ vgroup4 = VGroup(v10,v11,v12,v13,v14,v15,rect1,rect2,text3,text4,c2)
+ vgroup5 = VGroup(dot9,dot10,dot11,dot12,dot13,dot14,dot15)
+ text5 = TextMobject(r"$F$").scale(0.6).shift(3*UP+2*RIGHT)
+ dot9 = SmallDot(color=GREEN).shift(2*RIGHT+2*UP)
+ dot10 = SmallDot(color=GREEN).shift(2*RIGHT+1.5*UP)
+ dot11 = SmallDot(color=GREEN).shift(2*RIGHT+1*UP)
+ dot12 = SmallDot(color=GREEN).shift(2*RIGHT+0.5*UP)
+ dot13 = SmallDot(color=GREEN).shift(2*RIGHT)
+ dot14 = SmallDot(color=GREEN).shift(2*RIGHT+0.5*DOWN)
+ dot15 = SmallDot(color=GREEN).shift(2*RIGHT+1*DOWN)
+ f1 = TextMobject(r"${f_1}$").scale(0.5).shift(2.2*RIGHT+2*UP)
+ f2 = TextMobject(r"${f_2}$").scale(0.5).shift(2.2*RIGHT+1.5*UP)
+ f3 = TextMobject(r"${f_3}$").scale(0.5).shift(2.2*RIGHT+1*UP)
+ f4 = TextMobject(r"${f_4}$").scale(0.5).shift(2.2*RIGHT+0.5*UP)
+ f5 = TextMobject(r"${f_5}$").scale(0.5).shift(2.2*RIGHT)
+ f6 = TextMobject(r"${f_6}$").scale(0.5).shift(2.2*RIGHT+0.5*DOWN)
+ f7 = TextMobject(r"${f_7}$").scale(0.5).shift(2.2*RIGHT+1*DOWN)
+ vgroup6 = VGroup(f1,f2,f3,f4,f5,f6,f7)
+ arrow = Arrow(stroke_width=1.6).scale(1.5).shift(3*UP)
+ c3 = Ellipse(radius = 2,color=GREEN)
+ c3.rotate(np.pi/2)
+ c3.shift(2*RIGHT+0.6*UP)
+ c3.scale(2)
+ self.play(ShowCreation(v9))
+ self.wait(1.5)
+ self.play(ShowCreation(arrow),ShowCreation(colon),Transform(text2,text5),FadeOut(vgroup3),FadeOut(vgroup4),FadeOut(vgroup5))
+ self.play(ShowCreation(vgroup5),ShowCreation(vgroup6),ShowCreation(c3))
+ self.wait(0.7)
+ self.play(ShowCreation(vgroup3))
+ self.wait(3)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Dual_Basis_Example.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Dual_Basis_Example.py new file mode 100644 index 0000000..d79ec3e --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Dual_Basis_Example.py @@ -0,0 +1,97 @@ +from manimlib.imports import *
+import numpy as np
+
+class Dual_Basis(GraphScene):
+ CONFIG={
+ "x_min": -7,
+ "x_max": 7,
+ "y_min": -7,
+ "y_max": 7,
+ "graph_origin": ORIGIN,
+ "x_axis_label":"$X$",
+ "y_axis_label":"$Y$",
+ "x_labeled_nums": list(np.arange(-7, 8,1)),
+ "y_labeled_nums": list(np.arange(-7, 8,1)),
+ "x_axis_width": 10,
+ "y_axis_height": 10,
+ "x_tick_frequency":1,
+ "axes_color": GREY,
+ "area_opacity": 3,
+ "num_rects": 10,
+ }
+ def construct(self):
+ self.setup_axes(animate = True)
+ XD = self.x_axis_width/(self.x_max- self.x_min)
+ YD = self.y_axis_height/(self.y_max- self.y_min)
+ a1=2*XD*RIGHT+1*YD*UP
+ a2=3*XD*RIGHT+1*YD*UP
+ vec1=Vector(direction=a1,stroke_width=2).set_color(RED_E)
+ vec1.shift(self.graph_origin)
+ v1_label=TextMobject(r"$v_1$")
+ v1_label=(v1_label.shift(self.graph_origin+a1+0.1)).scale(.6)
+ self.play(ShowCreation(vec1),ShowCreation(v1_label))
+ text1=TextMobject(r"\text{$v_1$}",r"\text{$= (2,1)$}").scale(.6)
+ text1[0].set_color(RED_E)
+ text1.shift(5*LEFT+3.5*UP)
+ self.play(ShowCreation(text1))
+ self.wait(1.5)
+ vec2=Vector(direction=a2,stroke_width=2).set_color(YELLOW_E)
+ vec2.shift(self.graph_origin)
+ v2_label=TextMobject(r"$v_2$")
+ v2_label=(v2_label.shift(self.graph_origin+a2+0.1)).scale(.6)
+ self.play(ShowCreation(vec2),ShowCreation(v2_label))
+ text2=TextMobject(r"\text{$v_2$}",r"\text{$= (3,1)$}").scale(.6)
+ text2[0].set_color(YELLOW_E)
+ text2.shift(5*LEFT+3*UP)
+ self.play(ShowCreation(text2))
+ self.wait(1.5)
+ text3=TextMobject(r"\text{${T_2}$}",r"\text{$(v_1)$}",r"\text{$= 0$}").scale(.6)
+ text3[0].set_color(BLUE)
+ text3[1].set_color(RED_E)
+ text3.shift(4.94*LEFT+2.5*UP)
+ self.play(ShowCreation(text3))
+ self.wait(1.5)
+ text4=TextMobject(r"\text{${T_2}$}",r"\text{$= x - 2y$}").scale(.6)
+ text4[0].set_color(BLUE)
+ text4.shift(4.9*LEFT+2*UP)
+ self.play(ShowCreation(text4))
+ self.wait(1.5)
+ line1 = self.get_graph(lambda x : x/2, x_min = -5,x_max=5,color=BLUE)
+ v1_dual_label = TextMobject(r"${T_2}$").scale(.6).shift(3.9*RIGHT+1.85*UP)
+ self.play(ShowCreation(line1),ShowCreation(v1_dual_label))
+ self.wait(1.5)
+ text5=TextMobject(r"\text{${T_1}$}",r"\text{$(v_2)$}",r"\text{$= 0$}").scale(.6)
+ text5[1].set_color(YELLOW_E)
+ text5[0].set_color(PINK)
+ text5.shift(4.94*LEFT+1.5*UP)
+ self.play(ShowCreation(text5))
+ self.wait(1.5)
+ line2 = self.get_graph(lambda x : x/3, x_min = -5,x_max=5,color=PINK)
+ v2_dual_label = TextMobject(r"${T_1}$").scale(.6).shift(3.9*RIGHT+1.3*UP)
+ self.play(ShowCreation(line2),ShowCreation(v2_dual_label))
+ self.wait(1.5)
+ text6=TextMobject(r"\text{${T_1}$}",r"\text{$= - x + 3y$}").scale(.6)
+ text6[0].set_color(PINK)
+ text6.shift(4.76*LEFT+1*UP)
+ self.play(ShowCreation(text6))
+ self.wait(3)
+ text7 = TextMobject(r"\text{B =}",r"\text{$[$}",r"\text{$v_1,$}",r"\text{$v_2$}",r"\text{$]$}",r"\text{=}",r"\text{$[(2,1), (3,1)]$}").scale(0.6).shift(3*UP+4.5*LEFT)
+ text7[2].set_color(RED_E)
+ text7[3].set_color(YELLOW_E)
+ self.play(FadeOut(text1),FadeOut(text2),FadeOut(text3),FadeOut(text4),FadeOut(text5),FadeOut(text6))
+ self.play(ShowCreation(text7))
+ self.wait(0.7)
+ text8 = TextMobject(r"\text{B$^* =$}",r"\text{$[$}",r"\text{${T_1}$,}",r"\text{${T_2} $}",r"\text{$]$}",r"\text{=}",r"\text{$[-x + 3y, x - 2y]$}").scale(0.6).shift(2.3*UP+4.1*LEFT)
+ text8[3].set_color(BLUE)
+ text8[2].set_color(PINK)
+ self.play(ShowCreation(text8))
+ self.wait(3)
+
+
+
+
+
+
+
+
+
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Dual_of_a_Cube.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Dual_of_a_Cube.py new file mode 100644 index 0000000..a6f501e --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Dual_of_a_Cube.py @@ -0,0 +1,38 @@ +from manimlib.imports import *
+class Duality(ThreeDScene):
+
+
+ def construct(self):
+ axes = ThreeDAxes()
+ self.set_camera_orientation(phi = 65*DEGREES,theta =80*DEGREES)
+ self.begin_ambient_camera_rotation(rate=0.09)
+ cube = Cube(stroke_width=5,color=WHITE).scale(2)
+ cube.set_opacity(0.2)
+ self.play(ShowCreation(cube))
+ dot1= Dot(color=RED).scale(0.85).shift([2,0,0])
+ self.play(ShowCreation(dot1))
+ dot2= Dot(color=YELLOW).scale(0.85).shift([-2,0,0])
+ self.play(ShowCreation(dot2))
+ dot3= Dot(color=BLUE).scale(0.85).shift([0,-2,0])
+ self.play(ShowCreation(dot3))
+ dot4= Dot(color=GREEN).scale(0.85).shift([0,2,0])
+ self.play(ShowCreation(dot4))
+ dot5= Dot(color=ORANGE).scale(0.85).shift([0,0,2])
+ self.play(ShowCreation(dot5))
+ dot6= Dot(color=PINK).scale(0.85).shift([0,0,-2])
+ self.play(ShowCreation(dot6))
+ line1 = Line(start=[0,0,2],end=[2,0,0],stroke_width=2.5,color=BLACK)
+ line2 = Line(start=[0,0,2],end=[-2,0,0],stroke_width=2.5,color=BLACK)
+ line3 = Line(start=[0,0,2],end=[0,-2,0],stroke_width=2.5,color=BLACK)
+ line4 = Line(start=[0,0,2],end=[0,2,0],stroke_width=2.5,color=BLACK)
+ line5 = Line(start=[2,0,0],end=[0,0,-2],stroke_width=2.5,color=BLACK)
+ line6 = Line(start=[-2,0,0],end=[0,0,-2],stroke_width=2.5,color=BLACK)
+ line7 = Line(start=[0,-2,0],end=[0,0,-2],stroke_width=2.5,color=BLACK)
+ line8 = Line(start=[0,2,0],end=[0,0,-2],stroke_width=2.5,color=BLACK)
+ line9 = Line(start=[0,2,0],end=[-2,0,0],stroke_width=2.5,color=BLACK)
+ line10 = Line(start=[-2,0,0],end=[0,-2,0],stroke_width=2.5,color=BLACK)
+ line11 = Line(start=[0,-2,0],end=[2,0,0],stroke_width=2.5,color=BLACK)
+ line12 = Line(start=[2,0,0],end=[0,2,0],stroke_width=2.5,color=BLACK)
+ self.play(ShowCreation(line1),ShowCreation(line2),ShowCreation(line3),ShowCreation(line4),ShowCreation(line5),ShowCreation(line6),ShowCreation(line7),ShowCreation(line8),ShowCreation(line9),ShowCreation(line10),ShowCreation(line11),ShowCreation(line12))
+ self.wait(10)
+
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Duality_in_Sets.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Duality_in_Sets.py new file mode 100644 index 0000000..693017e --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Duality_in_Sets.py @@ -0,0 +1,25 @@ +from manimlib.imports import *
+import numpy as np
+class Duality_in_sets(Scene):
+ def construct(self):
+ circle1 = Circle(radius=0.4,color=BLACK).shift(2.3*LEFT)
+ circle1.set_fill(color=RED,opacity=200)
+ rect1=Rectangle(height=2,width=2,color=GREY).shift(2*LEFT)
+ rect1.set_fill(color=DARK_BLUE,opacity=1)
+ text1 = TextMobject("S").scale(0.7).shift(0.9*UP+0.7*LEFT)
+ text2 = TextMobject("X",color=BLACK,stroke_width=0.5).scale(0.5).shift(2.3*LEFT)
+ self.play(ShowCreation(rect1),ShowCreation(text1),ShowCreation(circle1),ShowCreation(text2))
+ circle2 = Circle(radius=0.4,color=BLACK).shift(1.7*RIGHT)
+ circle2.set_fill(color=BLACK,opacity=200)
+ rect2=Rectangle(height=2,width=2,color=GREY).shift(2*RIGHT)
+ rect2.set_fill(color=DARK_BLUE,opacity=1)
+ text3 = TextMobject("S").scale(0.7).shift(0.9*UP+3.3*RIGHT)
+ text4 = TextMobject(r"X$^c$",color=BLACK,stroke_width=0.2).scale(0.5).shift(2.55*RIGHT+0.6*UP)
+ text5 = TextMobject(r"\text{The subset}",r"\text{X$^c$}",r"\text{is the dual of subset}",r"\text{X}").scale(0.6).shift(2.7*UP+0.5*LEFT)
+ text5[1].set_color(GREY)
+ text5[3].set_color(GREY)
+ self.play(ShowCreation(rect2),ShowCreation(circle2),ShowCreation(text3),ShowCreation(text4))
+ self.wait(2)
+ self.play(ShowCreation(text5))
+ self.wait(3)
+
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Linear_Functional.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Linear_Functional.py new file mode 100644 index 0000000..6edc918 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Linear_Functional.py @@ -0,0 +1,29 @@ +from manimlib.imports import *
+import numpy as np
+class LinearFunctional(Scene):
+ def construct(self):
+ big_box=Rectangle().scale(2.7)
+ box = Rectangle(height=2,width=2,color=DARK_GREY).set_fill(color=PURPLE,opacity=350)
+ arrow1 = Arrow(color=RED).shift(1.8*LEFT+0.5*UP)
+ arrow2 = Arrow(color=RED).shift(1.8*LEFT+0.5*DOWN)
+ arrow3 = Arrow(color=GREEN).shift(0.5*UP+1.8*RIGHT)
+ arrow4 = Arrow(color=GREEN).shift(0.5*DOWN+1.8*RIGHT)
+ Linear = TextMobject("LINEAR",color=BLACK).scale(0.5).shift(0.2*UP)
+ Functional = TextMobject("FUNCTIONAL",color=BLACK).scale(0.5).shift(0.2*DOWN)
+ u = TextMobject(r"$\vec{u}$",color=YELLOW).scale(0.7).shift(2.8*LEFT+0.5*UP)
+ v = TextMobject(r"$\vec{v}$",color=YELLOW).scale(0.7).shift(2.8*LEFT+0.5*DOWN)
+ f1 = TextMobject(r"$f_1$",color=YELLOW).scale(0.7).shift(2.8*RIGHT+0.5*UP)
+ f2 = TextMobject(r"$f_2$",color=YELLOW).scale(0.7).shift(2.8*RIGHT+0.5*DOWN)
+ text = TextMobject(r"The Linear Functional is a function that takes $\vec{u}, \vec{v} \in$ V as inputs and gives the output $f_1, f_2\in$ F.").scale(0.55).shift(2*DOWN)
+ self.play(ShowCreation(big_box))
+ self.play(ShowCreation(box))
+ self.play(ShowCreation(Linear),ShowCreation(Functional))
+ self.wait(2)
+ self.play(ShowCreation(arrow1),ShowCreation(u))
+ self.play(ShowCreation(arrow3),ShowCreation(f1))
+ self.wait(0.7)
+ self.play(ShowCreation(arrow2),ShowCreation(v))
+ self.play(ShowCreation(arrow4),ShowCreation(f2))
+ self.wait(1)
+ self.play(ShowCreation(text))
+ self.wait(4)
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/gifs4/Basis_of_a_dual_vector_space.mp4 b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/gifs4/Basis_of_a_dual_vector_space.mp4 Binary files differnew file mode 100644 index 0000000..b96f541 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/gifs4/Basis_of_a_dual_vector_space.mp4 diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/gifs4/Dual_Basis_Example.mp4 b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/gifs4/Dual_Basis_Example.mp4 Binary files differnew file mode 100644 index 0000000..c93be25 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/gifs4/Dual_Basis_Example.mp4 diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/gifs4/Dual_of_a_Cube.mp4 b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/gifs4/Dual_of_a_Cube.mp4 Binary files differnew file mode 100644 index 0000000..ebfb564 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/gifs4/Dual_of_a_Cube.mp4 diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/gifs4/Duality_in_Sets.mp4 b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/gifs4/Duality_in_Sets.mp4 Binary files differnew file mode 100644 index 0000000..86cc693 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/gifs4/Duality_in_Sets.mp4 diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/gifs4/Linear_functional.mp4 b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/gifs4/Linear_functional.mp4 Binary files differnew file mode 100644 index 0000000..d41fada --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/gifs4/Linear_functional.mp4 diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Function_Scalar_Multiplication.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Function_Scalar_Multiplication.py new file mode 100644 index 0000000..ac74792 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Function_Scalar_Multiplication.py @@ -0,0 +1,62 @@ +from manimlib.imports import *
+from scipy import exp
+class FunctionScalarMultiplication(GraphScene):
+ CONFIG = {
+ "x_min" : -5,
+ "x_max" : 5,
+ "y_min" : -5,
+ "y_max" : 5,
+ "y_tick_frequency" : 1,
+ "x_tick_frequency" : 1,
+ "axes_color":LIGHT_GRAY,
+ "graph_origin": ORIGIN,
+ }
+ def construct(self):
+ self.setup_axes(animate = True)
+ curve1 = self.get_graph(lambda x : 1/3*(x**2)+1, x_min=-2,x_max=2.5,color=ORANGE)
+ curve2 = self.get_graph(lambda x : 2*(1/3*(x**2)+1), x_min=-2,x_max=2.5,color=GOLD)
+ fx= TextMobject("$f(x)$").scale(0.6).shift(1.25*UP + 2.2*LEFT)
+ gx= TextMobject("$2 \cdot f(x)$").scale(0.6).shift(0.5*UP + 2.1*LEFT)
+ f1 = self.get_vertical_line_to_graph(-2,curve1,color=YELLOW)
+ f2 = self.get_vertical_line_to_graph(-1.5,curve1,color=YELLOW)
+ f3 = self.get_vertical_line_to_graph(-1,curve1,color=YELLOW)
+ f4 = self.get_vertical_line_to_graph(-0.5,curve1,color=YELLOW)
+ f5 = self.get_vertical_line_to_graph(0,curve1,color=YELLOW)
+ f6 = self.get_vertical_line_to_graph(0.5,curve1,color=YELLOW)
+ f7 = self.get_vertical_line_to_graph(1,curve1,color=YELLOW)
+ f8 = self.get_vertical_line_to_graph(1.5,curve1,color=YELLOW)
+ f9 = self.get_vertical_line_to_graph(2,curve1,color=YELLOW)
+ f10 = self.get_vertical_line_to_graph(2.5,curve1,color=YELLOW)
+ self.play(ShowCreation(curve1),ShowCreation(fx))
+ self.wait(1.5)
+ self.play(ShowCreation(f1),ShowCreation(f2),ShowCreation(f3),ShowCreation(f4),ShowCreation(f5),ShowCreation(f6),ShowCreation(f7),ShowCreation(f8),ShowCreation(f9),ShowCreation(f10))
+ self.wait(1.7)
+ line1=Line(color=YELLOW).shift(5*LEFT+1.8*UP).scale(0.5)
+ line1.rotate(np.pi/2)
+ scalar = TextMobject("2 x").scale(0.65).shift(5.5*LEFT+1.9*UP)
+ equal = TextMobject("=").scale(0.65).shift(4.5*LEFT+1.9*UP)
+ line2=Line(color=BLUE).shift(4*LEFT+2.3*UP)
+ line2.rotate(np.pi/2)
+ self.play(ShowCreation(line1),ShowCreation(scalar),ShowCreation(equal),ShowCreation(line2))
+ g1 = self.get_vertical_line_to_graph(-2,curve2,color=BLUE)
+ g2 = self.get_vertical_line_to_graph(-1.5,curve2,color=BLUE)
+ g3 = self.get_vertical_line_to_graph(-1,curve2,color=BLUE)
+ g4 = self.get_vertical_line_to_graph(-0.5,curve2,color=BLUE)
+ g5 = self.get_vertical_line_to_graph(0,curve2,color=BLUE)
+ g6 = self.get_vertical_line_to_graph(0.5,curve2,color=BLUE)
+ g7 = self.get_vertical_line_to_graph(1,curve2,color=BLUE)
+ g8 = self.get_vertical_line_to_graph(1.5,curve2,color=BLUE)
+ g9 = self.get_vertical_line_to_graph(2,curve2,color=BLUE)
+ g10 = self.get_vertical_line_to_graph(2.5,curve2,color=BLUE)
+ self.play(ShowCreation(g1),ShowCreation(g2),ShowCreation(g3),ShowCreation(g4),ShowCreation(g5),ShowCreation(g6),ShowCreation(g7),ShowCreation(g8),ShowCreation(g9),ShowCreation(g10))
+ self.wait(2)
+ fx2=TextMobject("2$\cdot f(x)$").scale(0.6).shift(2.6*UP+2.3*LEFT)
+ self.play(ShowCreation(curve2),ShowCreation(fx2))
+ self.wait(1.5)
+ self.play(FadeOut(curve1),FadeOut(fx))
+ sc_mult=TextMobject("$(2 f(x)) = 2f(x)$",color=GOLD).scale(0.65).shift(0.65*UP + 5*LEFT)
+ rect = Rectangle(height=0.5,width=2)
+ rect.surround(sc_mult)
+ self.play(ShowCreation(sc_mult),ShowCreation(rect))
+ self.wait(3)
+
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Function_Space_Addition.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Function_Space_Addition.py new file mode 100644 index 0000000..5ce0e11 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Function_Space_Addition.py @@ -0,0 +1,89 @@ +from manimlib.imports import *
+from scipy import sin,exp
+class FunctionalVectorSpace(GraphScene):
+ CONFIG = {
+ "x_min" : -5,
+ "x_max" : 5,
+ "y_min" : -5,
+ "y_max" : 5,
+ "y_tick_frequency" : 1,
+ "x_tick_frequency" : 1,
+ "axes_color":LIGHT_GRAY,
+ "graph_origin": ORIGIN,
+
+
+ }
+ def construct(self):
+ self.setup_axes(animate = True)
+ curve1 = self.get_graph(lambda x : exp(x) + 0.5, x_min=-2,x_max=2.5,color=ORANGE)
+ curve2 = self.get_graph(lambda x : 1/3*(x**2)+1, x_min=-2,x_max=2.5,color=DARK_BLUE)
+ curve3 = self.get_graph(lambda x : 1/3*(x**2)+1 + exp(x) + 0.5, x_min=-2,x_max=2.5,color=BLACK)
+ fx= TextMobject("$f(x)$").scale(0.6).shift(1.25*UP + 2.1*LEFT)
+ gx= TextMobject("$g(x)$").scale(0.6).shift(0.5*UP + 2.1*LEFT)
+ f1 = self.get_vertical_line_to_graph(-2,curve1,color=YELLOW)
+ f2 = self.get_vertical_line_to_graph(-1.5,curve1,color=YELLOW)
+ f3 = self.get_vertical_line_to_graph(-1,curve1,color=YELLOW)
+ f4 = self.get_vertical_line_to_graph(-0.5,curve1,color=YELLOW)
+ f5 = self.get_vertical_line_to_graph(0,curve1,color=YELLOW)
+ f6 = self.get_vertical_line_to_graph(0.5,curve1,color=YELLOW)
+ f7 = self.get_vertical_line_to_graph(1,curve1,color=YELLOW)
+ f8 = self.get_vertical_line_to_graph(1.5,curve1,color=YELLOW)
+ f9 = self.get_vertical_line_to_graph(2,curve1,color=YELLOW)
+ f10 = self.get_vertical_line_to_graph(2.5,curve1,color=YELLOW)
+
+ self.play(ShowCreation(curve1),ShowCreation(gx))
+ self.wait(1.2)
+ self.play(ShowCreation(curve2),ShowCreation(fx))
+ self.wait(1.2)
+ self.play(ShowCreation(f1),ShowCreation(f2),ShowCreation(f3),ShowCreation(f4),ShowCreation(f5),ShowCreation(f6),ShowCreation(f7),ShowCreation(f8),ShowCreation(f9),ShowCreation(f10))
+ self.wait(1.7)
+ g1 = self.get_vertical_line_to_graph(-2,curve2,color=BLUE)
+ g2 = self.get_vertical_line_to_graph(-1.5,curve2,color=BLUE)
+ g3 = self.get_vertical_line_to_graph(-1,curve2,color=BLUE)
+ g4 = self.get_vertical_line_to_graph(-0.5,curve2,color=BLUE)
+ g5 = self.get_vertical_line_to_graph(0,curve2,color=BLUE)
+ g6 = self.get_vertical_line_to_graph(0.5,curve2,color=BLUE)
+ g7 = self.get_vertical_line_to_graph(1,curve2,color=BLUE)
+ g8 = self.get_vertical_line_to_graph(1.5,curve2,color=BLUE)
+ g9 = self.get_vertical_line_to_graph(2,curve2,color=BLUE)
+ g10 = self.get_vertical_line_to_graph(2.5,curve2,color=BLUE)
+ self.play(ShowCreation(g1),ShowCreation(g2),ShowCreation(g3),ShowCreation(g4),ShowCreation(g5),ShowCreation(g6),ShowCreation(g7),ShowCreation(g8),ShowCreation(g9),ShowCreation(g10))
+ line1=Line(color=BLUE).shift(5*LEFT+2.3*UP).scale(0.25)
+ line1.rotate(np.pi/2)
+ line2=Line(color=YELLOW).shift(6*LEFT+2.5*UP).scale(0.5)
+ line2.rotate(np.pi/2)
+ line3=Line(color=PURPLE_B).shift(4*LEFT+2.7*UP).scale(0.75)
+ line3.rotate(np.pi/2)
+ add=TextMobject("+").shift(2.4*UP+5.5*LEFT).scale(0.7)
+ equal=TextMobject("=").shift(2.4*UP+4.5*LEFT).scale(0.7)
+ self.play(ShowCreation(line2),ShowCreation(line1),ShowCreation(add),ShowCreation(equal),ShowCreation(line3))
+ self.wait(2)
+ self.play(FadeOut(curve1),FadeOut(curve2))
+ self.wait(3)
+ h1 = self.get_vertical_line_to_graph(-2,curve3,color=PURPLE_B)
+ h2 = self.get_vertical_line_to_graph(-1.5,curve3,color=PURPLE_B)
+ h3 = self.get_vertical_line_to_graph(-1,curve3,color=PURPLE_B)
+ h4 = self.get_vertical_line_to_graph(-0.5,curve3,color=PURPLE_B)
+ h5 = self.get_vertical_line_to_graph(0,curve3,color=PURPLE_B)
+ h6 = self.get_vertical_line_to_graph(0.5,curve3,color=PURPLE_B)
+ h7 = self.get_vertical_line_to_graph(1,curve3,color=PURPLE_B)
+ h8 = self.get_vertical_line_to_graph(1.5,curve3,color=PURPLE_B)
+ h9 = self.get_vertical_line_to_graph(2,curve3,color=PURPLE_B)
+ h10 = self.get_vertical_line_to_graph(2.5,curve3,color=PURPLE_B)
+
+ line1.shift(1*LEFT+0.9*UP)
+ equal.shift(0.3*LEFT+0.2*UP)
+ f=TextMobject("$f(x)$").scale(0.5).shift(5.6*LEFT+3.2*UP)
+ g=TextMobject("$g(x)$").scale(0.5).shift(5.6*LEFT+2.4*UP)
+ fg=TextMobject("$(f + g)(x)$").scale(0.5).shift(2.85*UP+3.3*LEFT)
+ self.play(FadeOut(add),ShowCreation(equal),ShowCreation(line1),ShowCreation(f),ShowCreation(g),ShowCreation(fg),FadeOut(fx),FadeOut(gx))
+ self.wait(1.7)
+ self.play(ShowCreation(h1),ShowCreation(h2),ShowCreation(h3),ShowCreation(h4),ShowCreation(h5),ShowCreation(h6),ShowCreation(h7),ShowCreation(h8),ShowCreation(h9),ShowCreation(h10))
+ curve3 = self.get_graph(lambda x : 1/3*(x**2)+1 + exp(x) + 0.5, x_min=-2,x_max=2.5,color=RED_A)
+ fgx=TextMobject("$(f + g)(x)$").scale(0.5).shift(1.65*UP+2.4*LEFT)
+ self.play(ShowCreation(curve3),ShowCreation(fgx))
+ sum=TextMobject("$(f + g)(x) = f(x) + g(x)$",color=GOLD).scale(0.65).shift(0.8*UP + 5*LEFT)
+ rect = Rectangle(height=0.5,width=2)
+ rect.surround(sum)
+ self.play(ShowCreation(sum),ShowCreation(rect))
+ self.wait(3)
\ No newline at end of file diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Integral_Properties.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Integral_Properties.py new file mode 100644 index 0000000..97c0e09 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Integral_Properties.py @@ -0,0 +1,77 @@ +from manimlib.imports import *
+from scipy import sin
+class Integral_Properties(GraphScene):
+ CONFIG = {
+ "x_min" : 0,
+ "x_max" : 5,
+ "y_min" : 0,
+ "y_max" : 6,
+ "y_tick_frequency" : 1,
+ "x_tick_frequency" : 1,
+ "axes_color":LIGHT_GRAY,
+ "x_labeled_nums" : list(range(6)),
+ "y_labeled_nums" : list(range(6))
+ }
+ def construct(self):
+ self.setup_axes(animate=False)
+ curve1 = self.get_graph(lambda x : sin(x), x_min=0,x_max=2.5,color=RED)
+ curve2 = self.get_graph(lambda x : x, x_min=0,x_max=2.5,color=DARK_BLUE)
+ curve3 = self.get_graph(lambda x : sin(x) + x, x_min=0,x_max=2.5,color=GREEN)
+ fx = TextMobject(r"$f(x)$").scale(0.5).shift(1*RIGHT+1.8*DOWN)
+ gx = TextMobject(r"$g(x)$").scale(0.5).shift(1*RIGHT)
+ sum = TextMobject(r"$f(x) + g(x)$").scale(0.5).shift(1.3*RIGHT+0.6*UP)
+ area1 = self.get_area(curve1,0,2.5)
+ area2 = self.get_area(curve2,0,2.5)
+ area3 = self.get_area(curve3,0,2.5)
+ area2.set_fill(color=PURPLE)
+ area3.set_fill(color=ORANGE)
+ text1=TextMobject(r"$\int_{0}^{2.5}$ f(x) dx = Area under the curve f(x)",color=BLUE_C).scale(0.7).shift(2.7*RIGHT+3*UP)
+ text2=TextMobject(r"$\int_{0}^{2.5}$ g(x) dx = Area under the curve g(x)",color=PURPLE_B).scale(0.7).shift(2.7*RIGHT+2.4*UP)
+ text3=TextMobject(r"Area under the curve f(x) + g(x) = $\int_{0}^{2.5} (f(x) + g(x)) dx$",color=ORANGE).scale(0.7).shift(2.7*RIGHT+1.8*UP)
+ text4=TextMobject(r"\text{$\int_{0}^{2.5}$ (f(x) + g(x)) dx}",r"\text{ = }",r"\text{ $\int_{0}^{2.5}$ f(x) dx }",r"\text{+}",r"\text{$\int_{0}^{2.5}$ g(x) dx}").scale(0.62).shift(2.7*RIGHT+2.7*UP)
+ text4[0].set_color(ORANGE)
+ text4[2].set_color(BLUE_C)
+ text4[4].set_color(PURPLE_B)
+ self.play(ShowCreation(curve1), ShowCreation(fx))
+ self.wait(1.2)
+ self.play(ShowCreation(curve2),ShowCreation(gx))
+ self.wait(1.2)
+ self.play(ShowCreation(area1))
+ self.play(ShowCreation(text1))
+ self.wait(1.5)
+ self.play(ShowCreation(area2))
+ self.play(ShowCreation(text2))
+ self.wait(1.5)
+ self.play(ShowCreation(curve3),ShowCreation(sum))
+ self.play(ShowCreation(area3))
+ self.play(ShowCreation(text3))
+ self.wait(2)
+ self.play(FadeOut(text1),FadeOut(text2),FadeOut(text3))
+ self.wait(1)
+ self.play(ShowCreation(text4))
+ self.wait(3)
+ self.play(FadeOut(curve1),FadeOut(curve2),FadeOut(area1),FadeOut(area2))
+ self.wait(1.5)
+ self.play(FadeOut(text4),FadeOut(area2),FadeOut(curve2),FadeOut(gx),FadeOut(curve3),FadeOut(sum),FadeOut(area3),ShowCreation(curve1),ShowCreation(fx))
+ self.wait(1.5)
+ self.play(ShowCreation(area1),ShowCreation(text1))
+ self.wait(1.5)
+ curve4 = self.get_graph(lambda x : 2*sin(x), x_min=0,x_max=2.5,color=RED)
+ area4 = self.get_area(curve4,0,2.5)
+ area4.set_fill(color=YELLOW)
+ fx2 = TextMobject(r"$2f(x)$").scale(0.7).shift(1*RIGHT+1.2*DOWN)
+ scalar_mul=TextMobject(r"$\int_{0}^{2.5} ( 2f(x) ) dx$ = 2 $\times$ Area under the curve f(x)",color=YELLOW).scale(0.7).shift(2.7*RIGHT+2.4*UP)
+ self.play(ShowCreation(curve4),ShowCreation(fx2))
+ self.wait(1)
+ self.play(ShowCreation(area4))
+ self.wait(2)
+ self.play(ShowCreation(scalar_mul))
+ self.wait(2)
+ text5=TextMobject(r"\text{$\int_{0}^{2.5}$ (2 f(x)) dx}",r"\text{ = }",r"\text{2 $\int_{0}^{2.5}$ f(x) dx }").scale(0.67).shift(2.7*RIGHT+2.7*UP)
+ text5[0].set_color(YELLOW)
+ text5[2].set_color(BLUE_C)
+ self.play(FadeOut(text1),FadeOut(scalar_mul))
+ self.wait(1)
+ self.play(ShowCreation(text5))
+ self.wait(3)
+
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/gifs3/Addition_of_Functions.gif b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/gifs3/Addition_of_Functions.gif Binary files differnew file mode 100644 index 0000000..6c42b74 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/gifs3/Addition_of_Functions.gif diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/gifs3/Function_Space_Example.gif b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/gifs3/Function_Space_Example.gif Binary files differnew file mode 100644 index 0000000..996a9de --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/gifs3/Function_Space_Example.gif diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/gifs3/Scalar_Multiplicaton_of_Functions.gif b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/gifs3/Scalar_Multiplicaton_of_Functions.gif Binary files differnew file mode 100644 index 0000000..8fff2c8 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/gifs3/Scalar_Multiplicaton_of_Functions.gif diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Conjugate_Symmetry_and_Positivity_of_Inner_Product.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Conjugate_Symmetry_and_Positivity_of_Inner_Product.py new file mode 100644 index 0000000..1d84842 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Conjugate_Symmetry_and_Positivity_of_Inner_Product.py @@ -0,0 +1,141 @@ +from manimlib.imports import *
+VECTORS = [[0, 2],
+ [1, 1],
+ [0, -2],
+ [1, -1],
+ [2, -2],
+ [2, 2],
+ [2, 2],
+ [-2, -1],
+ [3, 4]]
+class Scene1(LinearTransformationScene):
+ CONFIG = {
+ "include_background_plane": True,
+ "include_foreground_plane": False,
+ "show_coordinates": True,
+ "show_basis_vectors": False,
+ "basis_vector_stroke_width": 3,
+ }
+ def construct(self):
+ text1 = TextMobject(r"\text{$u$}",r"\text{ = $0 + 2i$, }",r"\text{$v$}",r"\text{ = $1 + i$}").scale(0.6).shift(3.5*UP+4*LEFT)
+ text1[0].set_color(YELLOW)
+ text1[2].set_color(RED)
+ text1.add_background_rectangle()
+ self.play(ShowCreation(text1))
+ text2 = TextMobject(r"\text{$\overline{u}$}",r"\text{ = $0 - 2i$, }",r"\text{$\overline{v}$}",r"\text{ = $1 - i$}").scale(0.6).shift(3*UP+4*LEFT)
+ text2[0].set_color(YELLOW)
+ text2[2].set_color(RED)
+ text2.add_background_rectangle()
+ self.play(ShowCreation(text2))
+ self.wait(2)
+ v1 = self.add_vector(VECTORS[0],color = YELLOW, stroke_width = 3.5)
+ u = TextMobject(r"$u$",color=YELLOW).shift(0.3*LEFT+2*UP).scale(0.6)
+ self.play(ShowCreation(u))
+ v1b = self.add_vector(VECTORS[2],color = YELLOW, stroke_width = 3.5)
+ ub = TextMobject(r"$\overline{u}$",color=YELLOW).shift(2*DOWN+0.3*LEFT).scale(0.6)
+ self.play(ShowCreation(ub))
+ self.wait(2)
+ v2 = self.add_vector(VECTORS[1],color = RED, stroke_width = 3.5)
+ v = TextMobject(r"$v$",color=RED).shift(1.2*RIGHT+1*UP).scale(0.6)
+ self.play(ShowCreation(v))
+ v2b = self.add_vector(VECTORS[3],color = RED, stroke_width = 3.5)
+ vb = TextMobject(r"$\overline{v}$",color=RED).shift(1.2*RIGHT+1*DOWN).scale(0.6)
+ self.play(ShowCreation(vb))
+ text3 = TextMobject(r"\text{$<u, v>$}",r"\text{ = }",r"\text{$\overline{u}$",r"\text{$\cdot$}",r"\text{$v$}",r"\text{ = }",r"\text{$2 - 2i$}").shift(2.5*UP+3.7*LEFT).scale(0.6)
+ text3[0].set_color(BLUE)
+ text3[2].set_color(YELLOW)
+ text3[4].set_color(RED)
+ text3.add_background_rectangle()
+ self.play(ShowCreation(text3))
+ self.wait(2)
+ text4 = TextMobject(r"\text{$<\overline{u, v}>$",r"\text{ = }",r"\text{$\overline{u}$",r"\text{$\cdot$}",r"\text{$v$}",r"\text{ = }",r"\text{$2 + 2i$}").shift(2*UP+3.7*LEFT).scale(0.6)
+ text4[0].set_color(BLUE)
+ text4[2].set_color(YELLOW)
+ text4[4].set_color(RED)
+ text4.add_background_rectangle()
+ line = Line(stroke_width = 1.5).scale(0.33).shift(2.2*UP+3.54*LEFT)
+ self.play(ShowCreation(text4),ShowCreation(line))
+ self.wait(2)
+ self.play(FadeOut(v1),FadeOut(v1b),FadeOut(v2),FadeOut(v2b),FadeOut(u),FadeOut(ub),FadeOut(v),FadeOut(vb))
+ v3 = self.add_vector(VECTORS[4],color = BLUE, stroke_width = 3.5)
+ uv = TextMobject(r"$\overline{u}\cdot v$",color=BLUE).shift(2.4*RIGHT+2.1*DOWN).scale(0.6)
+ self.play(ShowCreation(uv))
+ v3b = self.add_vector(VECTORS[5],color = BLUE, stroke_width = 3.5)
+ uvb = TextMobject(r"$\overline{\overline{u}\cdot v}$",color=BLUE).shift(2.4*RIGHT+2.1*UP).scale(0.6)
+ self.play(ShowCreation(uvb))
+ self.wait(2)
+ text5 = TextMobject(r"\text{$<v, u>$}",r"\text{ = }",r"\text{$\overline{v}$",r"\text{$\cdot$}",r"\text{$u$}",r"\text{ = }",r"\text{$2 + 2i$}").shift(1.5*UP+3.7*LEFT).scale(0.6)
+ text5[0].set_color(MAROON_B)
+ text5[2].set_color(RED)
+ text5[4].set_color(YELLOW)
+ text5.add_background_rectangle()
+ self.play(ShowCreation(text5))
+ self.wait(2)
+ v4 = self.add_vector(VECTORS[5],color = MAROON_B, stroke_width = 3.5)
+ vu = TextMobject(r"$\overline{v}\cdot u$",color=MAROON_B).shift(1.3*RIGHT+2.1*UP).scale(0.6)
+ self.play(ShowCreation(vu))
+ self.play(FadeOut(uvb))
+ self.wait(2)
+ text6 = TextMobject(r"\text{$<\overline{u, v}>$",r"\text{ = }",r"\text{$<v, u>$").scale(0.6).shift(0.8*UP+4.2*LEFT)
+ text6[0].set_color(BLUE)
+ text6[2].set_color(MAROON_B)
+ text6.add_background_rectangle()
+ self.play(ShowCreation(text6))
+ rect = Rectangle(height = 0.7,width = 3.5)
+ rect.surround(text6)
+ self.play(ShowCreation(rect))
+ self.wait(3)
+ self.play(FadeOut(line),FadeOut(text1),FadeOut(text2),FadeOut(text3),FadeOut(text4),FadeOut(text5),FadeOut(text6),FadeOut(v4),FadeOut(vu),FadeOut(v3),FadeOut(uv),FadeOut(rect),FadeOut(v3b))
+
+ text7 = TextMobject(r"\text{$u$}",r"\text{ = $(1 + i) > 0$}").scale(0.6).shift(3.5*UP+4.5*LEFT)
+ text7[0].set_color(YELLOW)
+ text7.add_background_rectangle()
+ self.play(ShowCreation(text7))
+ v5 = self.add_vector(VECTORS[1],color = YELLOW, stroke_width = 3.5)
+ u = TextMobject(r"$u$",color=YELLOW).shift(1.2*RIGHT+1*UP).scale(0.6)
+ self.play(ShowCreation(u))
+ self.wait(1.5)
+ text8 = TextMobject(r"\text{$<u, u>$}",r"\text{ = $(0 + 2i) > 0$ }").scale(0.6).shift(2.7*UP+4*LEFT)
+ text8[0].set_color(GREEN)
+ text8.add_background_rectangle()
+ rect1 = Rectangle(height = 0.55, width = 3.3)
+ rect1.surround(text8)
+ self.play(ShowCreation(text8),ShowCreation(rect1))
+ self.wait(2)
+ v6 = self.add_vector(VECTORS[0],color = GREEN, stroke_width = 3.5)
+ uu = TextMobject(r"$<u, u>$",color=GREEN).shift(0.8*LEFT+1.9*UP).scale(0.6)
+ self.play(ShowCreation(uu))
+ text9 = TextMobject(r"\text{$v$}",r"\text{ = $(-2 - i) < 0$}").scale(0.6).shift(1.5*UP+4.4*LEFT)
+ text9[0].set_color(RED)
+ text9.add_background_rectangle()
+ self.play(ShowCreation(text9))
+ self.wait(1.5)
+ v7 = self.add_vector(VECTORS[7],color = RED, stroke_width = 3.5)
+ v = TextMobject(r"$v$",color=RED).shift(2.2*LEFT+1*DOWN).scale(0.6)
+ self.play(ShowCreation(v))
+ self.wait(1.5)
+ text10 = TextMobject(r"\text{$<v, v>$}",r"\text{ = $(3 + 4i) > 0$ }").scale(0.6).shift(0.7*UP+4*LEFT)
+ text10[0].set_color(BLUE)
+ text10.add_background_rectangle()
+ rect2 = Rectangle(height=0.55,width=3.3)
+ rect2.surround(text10)
+ self.play(ShowCreation(text10),ShowCreation(rect2))
+ self.wait(2)
+ v8 = self.add_vector(VECTORS[8],color = BLUE, stroke_width = 3.5)
+ vv = TextMobject(r"$<v, v>$",color=BLUE).shift(2.1*RIGHT+3.8*UP).scale(0.6)
+ self.play(ShowCreation(vv))
+ self.wait(4)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Inner_Product_Example.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Inner_Product_Example.py new file mode 100644 index 0000000..97b9696 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Inner_Product_Example.py @@ -0,0 +1,182 @@ +from manimlib.imports import *
+from scipy import sin,cos
+class Inner_Product_Space_Example(GraphScene):
+ CONFIG = {
+ "x_min" : 0,
+ "x_max" : 5,
+ "y_min" : 0,
+ "y_max" : 6,
+ "y_tick_frequency" : 1,
+ "x_tick_frequency" : 1,
+ "axes_color":LIGHT_GRAY,
+ "x_labeled_nums" : list(range(6)),
+ "y_labeled_nums" : list(range(6))
+ }
+ def construct(self):
+ self.setup_axes(animate=True)
+ text = TextMobject(r"$f(x), g(x), h(x) \in C[0, 2]$",color=GOLD).scale(0.5).shift(3.5*UP+5.5*LEFT)
+ fx = TextMobject(r"$f(x)$ = sin(x)",color=GOLD).scale(0.5).shift(3*UP+6*LEFT)
+ gx = TextMobject(r"$g(x)$ = x",color=GOLD).scale(0.5).shift(2.5*UP+6.25*LEFT)
+ hx = TextMobject(r"$h(x)$ = 1.4",color=GOLD).scale(0.5).shift(2*UP+6.2*LEFT)
+
+ curve1 = self.get_graph(lambda x : sin(x), x_min=0,x_max=2,color=RED)
+ curve2 = self.get_graph(lambda x : x, x_min=0,x_max=2,color=DARK_BLUE)
+ curve3 = self.get_graph(lambda x : 1.4, x_min=0,x_max=2,color=GREEN)
+ text1 = TextMobject(r"$f(x)$").scale(0.5).shift(1.7*DOWN)
+
+ text2 = TextMobject(r"$g(x)$").scale(0.5).shift(0.34*DOWN)
+ text3 = TextMobject(r"$h(x)$").scale(0.5).shift(1.1*DOWN)
+
+
+ self.play(ShowCreation(text))
+ self.play(ShowCreation(curve1),ShowCreation(text1),ShowCreation(fx))
+ self.wait(1)
+ self.play(ShowCreation(curve2),ShowCreation(text2),ShowCreation(gx))
+ self.wait(1)
+ self.play(ShowCreation(curve3),ShowCreation(text3),ShowCreation(hx))
+ self.wait(2)
+ curve4 = self.get_graph(lambda x : sin(x) + x, x_min=0,x_max=2,color=YELLOW)
+ text4 = TextMobject(r"$f(x) + g(x)$").scale(0.5).shift(0.5*UP+0.5*RIGHT)
+ self.wait(1.5)
+
+ self.play(ShowCreation(curve4),ShowCreation(text4),FadeOut(curve2),FadeOut(text2),FadeOut(curve1),FadeOut(text1))
+ self.wait(1.5)
+ text5 = TextMobject(r"\text{$<f(x) + g(x), h(x)>$ = ",r"\text{$\int_{0}^{2} (f(x) + g(x))h(x)$ $dx$}",r"\text{= 4.78}").scale(0.57).shift(3.3*RIGHT+3.5*UP)
+ text5[1].set_color(ORANGE)
+ self.play(ShowCreation(text5))
+
+ curve5 = self.get_graph(lambda x : (sin(x) + x)*1.6, x_min=0,x_max=2,color=ORANGE)
+ text6 = TextMobject(r"$(f(x) + g(x))\cdot h(x)$").scale(0.5).shift(2.2*UP+0.72*RIGHT)
+ area1 = self.get_area(curve5,0,2)
+ area1.set_color(ORANGE)
+ self.wait(1)
+ self.play(FadeOut(curve4),FadeOut(text4),FadeOut(curve3),FadeOut(text3),ShowCreation(curve5),ShowCreation(text6),ShowCreation(area1))
+ self.wait(2)
+ text7 = TextMobject(r"\text{$<f(x), h(x)>$ = ",r"\text{$\int_{0}^{2} (f(x)h(x)$ $dx$}",r"\text{= 1.98}").scale(0.57).shift(4.5*RIGHT+3*UP)
+ text7[1].set_color(BLUE)
+ self.play(ShowCreation(text7))
+ self.wait(1.5)
+ curve6 = self.get_graph(lambda x : (sin(x))*1.6, x_min=0,x_max=2,color=BLUE)
+ text8 = TextMobject(r"$f(x)\cdot h(x)$").scale(0.5).shift(0.9*DOWN+0.3*RIGHT)
+ area2 = self.get_area(curve6,0,2)
+ self.play(ShowCreation(curve6),ShowCreation(text8),ShowCreation(area2))
+ self.wait(1.5)
+ text9 = TextMobject(r"\text{$<g(x), h(x)>$ = ",r"\text{$\int_{0}^{2} (g(x)h(x)$ $dx$}",r"\text{= 2.8}").scale(0.57).shift(4.5*RIGHT+2.5*UP)
+ text9[1].set_color(MAROON_B)
+ self.play(ShowCreation(text9))
+ self.wait(1.5)
+ curve7 = self.get_graph(lambda x : x*1.6, x_min=0,x_max=2,color=MAROON_B)
+ text10 = TextMobject(r"$g(x)\cdot h(x)$").scale(0.5).shift(0.3*RIGHT+0.78*UP)
+ area3 = self.get_area(curve7,0,2)
+ area3.set_color(MAROON_B)
+ self.play(ShowCreation(curve7),ShowCreation(text10),ShowCreation(area3))
+ self.wait(2.6)
+ curve8 = self.get_graph(lambda x : (sin(x))*1.6 + x*1.6, x_min=0,x_max=2,color=RED_C)
+ area4 = self.get_area(curve8,0,2)
+ area4.set_color(RED_C)
+ text11 = TextMobject(r"$f(x)h(x) + g(x)h(x)$").scale(0.5).shift(2.2*UP + 0.76*RIGHT)
+ self.play(FadeOut(curve6),FadeOut(text8),FadeOut(curve7),FadeOut(text10),FadeOut(area2),FadeOut(area3),ShowCreation(curve8),ShowCreation(area4))
+ self.wait(1)
+ self.play(Transform(text6,text11))
+ self.wait(1.7)
+ text12 = TextMobject(r"$<f(x) + g(x), h(x)>$ = $<f(x), h(x)>$ + $<g(x), h(x)>$").scale(0.465).shift(0.7*UP+4*RIGHT)
+ rect1 = Rectangle(height=0.5)
+ rect1.surround(text12)
+ self.play(ShowCreation(text12),ShowCreation(rect1))
+ self.wait(3)
+ self.play(FadeOut(text6),FadeOut(text5),FadeOut(text7),FadeOut(text9),FadeOut(text12),FadeOut(rect1),FadeOut(curve8),FadeOut(area4),FadeOut(text11),FadeOut(curve5),FadeOut(area1))
+
+ curve2.set_color(ORANGE)
+ self.play(ShowCreation(curve1),ShowCreation(text1))
+ self.wait(1)
+ self.play(ShowCreation(curve2),ShowCreation(text2))
+ self.wait(2)
+ curve9 = self.get_graph(lambda x : 2*sin(x), x_min=0,x_max=2,color=GREEN)
+ text13 = TextMobject(r"$2f(x)$").scale(0.5).shift(0.75*DOWN)
+ self.play(Transform(curve1,curve9),Transform(text1,text13))
+ self.wait(1.5)
+
+ text14 = TextMobject(r"\text{$<2f(x), g(x)>$ = ",r"\text{$\int_{0}^{2} (2f(x))g(x)$ $dx$}",r"\text{= 3.48}").scale(0.57).shift(4*RIGHT+3.5*UP)
+ text14[1].set_color(YELLOW)
+ self.play(ShowCreation(text14))
+ self.wait(2.2)
+ curve10 = self.get_graph(lambda x : 2*sin(x)*x, x_min=0,x_max=2,color=YELLOW)
+ text15 = TextMobject(r"$2f(x)\cdot g(x)$").scale(0.5).shift(0.35*RIGHT+1.03*UP)
+ area5 = self.get_area(curve10,0,2)
+ area5.set_color(YELLOW)
+ self.play(ShowCreation(area5),ShowCreation(curve10),ShowCreation(text15),FadeOut(curve1),FadeOut(text1),FadeOut(curve2),FadeOut(text2))
+ self.wait(2)
+ text16 = TextMobject(r"\text{$<f(x), g(x)>$ = ",r"\text{$\int_{0}^{2} f(x)g(x)$ $dx$}",r"\text{= 1.74}").scale(0.57).shift(3.8*RIGHT+2.9*UP)
+ text16[1].set_color(TEAL)
+ self.play(ShowCreation(text16))
+ self.wait(1.7)
+ curve11 = self.get_graph(lambda x : sin(x)*x, x_min=0,x_max=2,color=TEAL)
+ area6 = self.get_area(curve11,0,2)
+ area6.set_color(TEAL)
+ text17 = TextMobject(r"$f(x)\cdot g(x)$").scale(0.5).shift(0.4*RIGHT+0.7*DOWN)
+ self.play(ShowCreation(curve11),ShowCreation(text17),ShowCreation(area6))
+ self.wait(2)
+
+ text18 = TextMobject(r"\text{$2 <f(x), g(x)>$ = ",r"\text{$2 \int_{0}^{2} f(x)g(x)$ $dx$}",r"\text{= 3.48}").scale(0.57).shift(4*RIGHT+2.3*UP)
+ text18[1].set_color(DARK_BLUE)
+ self.play(ShowCreation(text18))
+ self.wait(2)
+ curve12 = self.get_graph(lambda x : 2*sin(x)*x, x_min=0,x_max=2,color=DARK_BLUE)
+ area7 = self.get_area(curve12,0,2)
+ area7.set_color(DARK_BLUE)
+ text19 = TextMobject(r"= $2( f(x)\cdot g(x) )$").scale(0.5).shift(1.89*RIGHT+1.03*UP)
+ self.play(ShowCreation(curve12),ShowCreation(area7),ShowCreation(text19),FadeOut(text17),FadeOut(area6),FadeOut(curve11))
+
+ self.wait(2.5)
+ text20 = TextMobject(r"$<2f(x), g(x)>$ = $2<f(x), g(x)>$").scale(0.57).shift(0.6*DOWN+4*RIGHT)
+ rect2 = Rectangle(height=0.5)
+ rect2.surround(text20)
+ self.play(ShowCreation(text20),ShowCreation(rect2))
+ self.wait(3)
+
+ self.play(FadeOut(text14),FadeOut(text15),FadeOut(text19),FadeOut(text16),FadeOut(text18),FadeOut(rect2),FadeOut(curve10),FadeOut(area5),FadeOut(curve12),FadeOut(area7),FadeOut(text20))
+ curve1 = self.get_graph(lambda x : sin(x), x_min=0,x_max=2,color=YELLOW)
+ text1 = TextMobject(r"$f(x)$").scale(0.5).shift(1.77*DOWN)
+ self.play(ShowCreation(curve1),ShowCreation(text1))
+ self.wait(1.5)
+ self.play(ShowCreation(curve2),ShowCreation(text2))
+ self.wait(1.7)
+ text21 = TextMobject(r"\text{$<f(x), g(x)>$ = ",r"\text{$\int_{0}^{2} f(x)g(x)$ $dx$}",r"\text{= 1.74}").scale(0.57).shift(3.5*RIGHT+3*UP)
+ text21[1].set_color(GREEN)
+ self.play(ShowCreation(text21))
+ self.wait(2)
+ curve13 = self.get_graph(lambda x : sin(x)*x, x_min=0,x_max=2,color=GREEN)
+ area8 = self.get_area(curve13,0,2)
+ area8.set_color(GREEN)
+ text22 = TextMobject(r"$f(x)\cdot g(x)$").scale(0.5).shift(0.32*RIGHT+0.7*DOWN)
+ self.play(ShowCreation(curve13),ShowCreation(area8),ShowCreation(text22),FadeOut(curve1),FadeOut(text1),FadeOut(curve2),FadeOut(text2))
+ self.wait(2.2)
+ curve14 = self.get_graph(lambda x : sin(x)*x, x_min=0,x_max=2,color=RED)
+ area9 = self.get_area(curve14,0,2)
+ area9.set_color(RED)
+ self.play(ShowCreation(curve14),ShowCreation(area9))
+ text23 = TextMobject(r"= $\overline{f(x)\cdot g(x)}$").scale(0.5).shift(0.7*DOWN+1.7*RIGHT)
+ self.play(ShowCreation(text23))
+ self.wait(2)
+ text24 = TextMobject(r"For all the real functions").scale(0.5).shift(2*RIGHT+2*UP)
+ text25 = TextMobject(r"$<\overline{f(x), g(x)}>$ = $<f(x), g(x)>$").scale(0.5).shift(2*RIGHT+1.4*UP)
+ rect3 = Rectangle(height=0.7)
+ rect3.surround(text25)
+ self.play(ShowCreation(text24),ShowCreation(text25),ShowCreation(rect3))
+ self.wait(3)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Inner_product.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Inner_product.py new file mode 100644 index 0000000..e1344f7 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Inner_product.py @@ -0,0 +1,28 @@ +from manimlib.imports import *
+import numpy as np
+class InnerProduct(Scene):
+ def construct(self):
+ big_box=Rectangle().scale(2.7)
+ box = Rectangle(height=2,width=2,color=DARK_GREY).set_fill(color=BLUE_B,opacity=350)
+ arrow1 = Arrow(color=RED).shift(1.8*LEFT+0.5*UP)
+ arrow2 = Arrow(color=RED).shift(1.8*LEFT+0.5*DOWN)
+ arrow3 = Arrow(color=GREEN).shift(1.8*RIGHT)
+ Inner = TextMobject("INNER",color=BLACK).scale(0.65).shift(0.2*UP)
+ Product = TextMobject("PRODUCT",color=BLACK).scale(0.65).shift(0.2*DOWN)
+ u = TextMobject(r"$\vec{u}$",color=YELLOW).scale(0.7).shift(2.8*LEFT+0.5*UP)
+ v = TextMobject(r"$\vec{v}$",color=YELLOW).scale(0.7).shift(2.8*LEFT+0.5*DOWN)
+ scalar = TextMobject("(Scalar)").scale(0.55).shift(3.3*RIGHT+0.4*DOWN)
+ u.v = TextMobject(r"$<\vec{u}, \vec{v}>$",color=YELLOW).scale(0.7).shift(3.3*RIGHT)
+ V = TextMobject("V").scale(0.7).shift(2*UP+3.8*RIGHT)
+ text = TextMobject(r"The Inner Product is an operation that takes $\vec{u}, \vec{v} \in$ V as inputs and gives a scalar as an output.").scale(0.55).shift(2*DOWN)
+ self.play(ShowCreation(big_box),ShowCreation(V))
+ self.play(ShowCreation(box))
+ self.play(ShowCreation(Product),ShowCreation(Inner))
+ self.wait(2)
+ self.play(ShowCreation(arrow1),ShowCreation(u))
+ self.play(ShowCreation(arrow2),ShowCreation(v))
+ self.wait(1.5)
+ self.play(ShowCreation(arrow3),ShowCreation(scalar),ShowCreation(u.v))
+ self.wait(1)
+ self.play(ShowCreation(text))
+ self.wait(10)
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Instances_of_Topological_Spaces.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Instances_of_Topological_Spaces.py new file mode 100644 index 0000000..2b3cf14 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Instances_of_Topological_Spaces.py @@ -0,0 +1,36 @@ +from manimlib.imports import *
+class InnerProduct(Scene):
+ def construct(self):
+
+ circle1 = Circle(color=DARK_GREY).scale(3.7)
+ circle1.set_fill(color=GOLD,opacity=350)
+ square1 = Square(color=DARK_GREY).scale(2.5)
+ square1.set_fill(color=DARK_BLUE,opacity=350)
+ square2=Square(color=DARK_GREY).scale(1.6).shift(0.2*DOWN)
+ square2.rotate(np.pi/4).set_fill(color=YELLOW,opacity=350)
+ square3 = Square(color=DARK_GREY).scale(1).shift(0.3*DOWN)
+ square3.set_fill(color=BLACK,opacity=350)
+ circle2 = Circle(color=DARK_GREY).scale(0.5).shift(0.45*DOWN)
+ circle2.set_fill(color=MAROON_A,opacity=350)
+ text1 = TextMobject("Topological Spaces",color=BLACK).scale(0.5).shift(3*UP)
+ text2 = TextMobject("Metric Spaces",color=BLACK).scale(0.5).shift(2.2*UP)
+ text3 = TextMobject("Normed",color=BLACK).scale(0.44).shift(1.4*UP)
+ text4 = TextMobject("Vector Spaces",color=BLACK).scale(0.44).shift(1.17*UP)
+ text5 = TextMobject("Inner Product",color=WHITE).scale(0.37).shift(0.5*UP)
+ text6 = TextMobject("Spaces",color=WHITE).scale(0.37).shift(0.27*UP)
+ text7 = TextMobject("Dot",color=BLACK).scale(0.37).shift(0.3*DOWN)
+ text8 = TextMobject("Product",color=BLACK).scale(0.32).shift(0.5*DOWN)
+
+
+
+
+ self.play(ShowCreation(circle1),ShowCreation(text1))
+ self.wait(1.5)
+ self.play(ShowCreation(square1),ShowCreation(text2))
+ self.wait(1.5)
+ self.play(ShowCreation(square2),ShowCreation(text3),ShowCreation(text4))
+ self.wait(1.5)
+ self.play(ShowCreation(square3),ShowCreation(text5),ShowCreation(text6))
+ self.wait(1.5)
+ self.play(ShowCreation(circle2),ShowCreation(text7),ShowCreation(text8))
+ self.wait(4)
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Interpretation_of_Norm_as_Length.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Interpretation_of_Norm_as_Length.py new file mode 100644 index 0000000..23c568d --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Interpretation_of_Norm_as_Length.py @@ -0,0 +1,42 @@ +from manimlib.imports import *
+import numpy as np
+class Interpretation_of_norm_as_length(GraphScene):
+ CONFIG = {
+ "x_min" : 0,
+ "x_max" : 5,
+ "y_min" : 0,
+ "y_max" : 5,
+ "y_tick_frequency" : 1,
+ "x_tick_frequency" : 1,
+ "axes_color":LIGHT_GRAY,
+ "x_axis_width": 5,
+ "y_axis_height":5,
+ "graph_origin" : ORIGIN + 2*DOWN + 2*LEFT,
+ "enclude_zero_label": False
+
+ }
+ def construct(self):
+ self.setup_axes(animate=False)
+ dot = Dot().scale(0.5)
+ self.play(ShowCreation(dot))
+ origin = TextMobject(r"(0, 0)").scale(0.5).shift(2.5*LEFT+2.2*DOWN)
+ line1 = self.get_graph(lambda x : x, x_min=0,x_max=2,color=WHITE)
+ line2 = Line().rotate(np.pi/2).shift(1*DOWN)
+ text1 = TextMobject(r"$(v_1, 0)$").scale(0.5).shift(2.2*DOWN+0.2*RIGHT)
+ text2 = TextMobject(r"$(0, v_2)$").scale(0.5).shift(2.5*LEFT)
+ text3 = TextMobject(r"$(v_1, v_2)$").scale(0.5).shift(0.5*RIGHT)
+ text4 = TextMobject(r"$| v_1 |$",color=RED_B).scale(0.5).shift(1*LEFT+2.3*DOWN)
+ text5 = TextMobject(r"$| v_2 |$",color=RED_B).scale(0.5).shift(0.3*RIGHT+1*DOWN)
+ text6 = TextMobject(r"$\sqrt{{v_1}^2 + {v_2}^2}$",color=RED_B).scale(0.5).rotate(np.pi/4).shift(1.3*LEFT+1*DOWN)
+ line3 = Line(color=YELLOW).shift(1*LEFT+2*DOWN)
+ self.play(ShowCreation(line1),ShowCreation(line2),ShowCreation(text1),ShowCreation(text2),ShowCreation(text3),ShowCreation(origin))
+ self.wait(1.5)
+ self.play(ShowCreation(line3),ShowCreation(text4))
+ self.wait(1.5)
+ line2 = Line(color=YELLOW).rotate(np.pi/2).shift(1*DOWN)
+ self.play(ShowCreation(line2),ShowCreation(text5))
+ self.wait(1.5)
+ line1 = self.get_graph(lambda x : x, x_min=0,x_max=2,color=YELLOW)
+ self.play(ShowCreation(text6),ShowCreation(line1))
+ self.wait(4)
+
\ No newline at end of file diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Linearity_of_Inner_Product.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Linearity_of_Inner_Product.py new file mode 100644 index 0000000..e1344f7 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Linearity_of_Inner_Product.py @@ -0,0 +1,28 @@ +from manimlib.imports import *
+import numpy as np
+class InnerProduct(Scene):
+ def construct(self):
+ big_box=Rectangle().scale(2.7)
+ box = Rectangle(height=2,width=2,color=DARK_GREY).set_fill(color=BLUE_B,opacity=350)
+ arrow1 = Arrow(color=RED).shift(1.8*LEFT+0.5*UP)
+ arrow2 = Arrow(color=RED).shift(1.8*LEFT+0.5*DOWN)
+ arrow3 = Arrow(color=GREEN).shift(1.8*RIGHT)
+ Inner = TextMobject("INNER",color=BLACK).scale(0.65).shift(0.2*UP)
+ Product = TextMobject("PRODUCT",color=BLACK).scale(0.65).shift(0.2*DOWN)
+ u = TextMobject(r"$\vec{u}$",color=YELLOW).scale(0.7).shift(2.8*LEFT+0.5*UP)
+ v = TextMobject(r"$\vec{v}$",color=YELLOW).scale(0.7).shift(2.8*LEFT+0.5*DOWN)
+ scalar = TextMobject("(Scalar)").scale(0.55).shift(3.3*RIGHT+0.4*DOWN)
+ u.v = TextMobject(r"$<\vec{u}, \vec{v}>$",color=YELLOW).scale(0.7).shift(3.3*RIGHT)
+ V = TextMobject("V").scale(0.7).shift(2*UP+3.8*RIGHT)
+ text = TextMobject(r"The Inner Product is an operation that takes $\vec{u}, \vec{v} \in$ V as inputs and gives a scalar as an output.").scale(0.55).shift(2*DOWN)
+ self.play(ShowCreation(big_box),ShowCreation(V))
+ self.play(ShowCreation(box))
+ self.play(ShowCreation(Product),ShowCreation(Inner))
+ self.wait(2)
+ self.play(ShowCreation(arrow1),ShowCreation(u))
+ self.play(ShowCreation(arrow2),ShowCreation(v))
+ self.wait(1.5)
+ self.play(ShowCreation(arrow3),ShowCreation(scalar),ShowCreation(u.v))
+ self.wait(1)
+ self.play(ShowCreation(text))
+ self.wait(10)
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Projection-in-3D-space.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Projection-in-3D-space.py new file mode 100644 index 0000000..e99202f --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Projection-in-3D-space.py @@ -0,0 +1,31 @@ +from manimlib.imports import *
+class ThreeDSpace(ThreeDScene):
+
+ def construct(self):
+ axes = ThreeDAxes()
+ axes.set_stroke(width=1,color=GOLD)
+ self.add(axes)
+ self.set_camera_orientation(phi = 70*DEGREES,theta =110*DEGREES)
+ line1 = Line(color = YELLOW,opacity=350,start = ORIGIN,end = [0.5,2,2])
+ self.play(ShowCreation(line1))
+ self.wait(1)
+ line2 = Line(color = BLUE,opacity=350,start = ORIGIN,end = [2,2,-1])
+ self.play(ShowCreation(line2))
+ self.wait(1)
+ line3 = Line(opacity=350,start=axes.c2p(0.5,2,2) ,end=axes.c2p(0.5,1.14,-0.17))
+ self.play(ShowCreation(line3))
+ self.wait(1)
+ line4 = Line(color=RED,opacity=350,start = ORIGIN,end=[0.5,1.14,-0.145])
+ self.play(ShowCreation(line4))
+ self.wait(2)
+ text1 = TextMobject(r"\text{Projection on }",r"\text{$\vec{v}$}",r"\text{ onto }",r"\text{$\vec{u}$}",r"\text{ = }",r"\text{$ |\vec{v}|cos\theta$,}").scale(0.6).shift(4*LEFT+3.5*UP)
+ text1[1].set_color(YELLOW)
+ text1[3].set_color(BLUE)
+ text1[5].set_color(RED)
+ text2 = TextMobject(r"\text{where $\theta$ is the angle between the vectors.}").scale(0.6).shift(4*LEFT+3*UP)
+ self.add_fixed_in_frame_mobjects(text1)
+ self.add_fixed_in_frame_mobjects(text2)
+ self.play(ShowCreation(text1),ShowCreation(text2))
+ self.wait(3)
+
+
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Vector_Projection.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Vector_Projection.py new file mode 100644 index 0000000..b1724c1 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Vector_Projection.py @@ -0,0 +1,43 @@ +from manimlib.imports import *
+VECTORS = [[3,1],
+ [1,3],
+ [1.96,0.65]]
+class Projection(LinearTransformationScene):
+ CONFIG = {
+ "include_background_plane": True,
+ "include_foreground_plane": False,
+ "show_coordinates": False,
+ "show_basis_vectors": False,
+ "basis_vector_stroke_width": 3,
+ }
+ def construct(self):
+ v1 = self.add_vector(VECTORS[0],color = YELLOW, stroke_width = 3.5)
+ u = TextMobject(r"$\vec{u}$",color=YELLOW).scale(0.65).shift(3.2*RIGHT,1*UP)
+ self.play(ShowCreation(u))
+ self.wait(0.7)
+ v2 = self.add_vector(VECTORS[1],color = RED, stroke_width = 3.5)
+ v = TextMobject(r"$\vec{v}$",color=RED).scale(0.65).shift(1.2*RIGHT,3*UP)
+ self.play(ShowCreation(v))
+ self.wait(0.7)
+ angle = Arc(radius=0.6).scale(0.5)
+ theta = TextMobject(r"$\theta$").scale(0.65).shift(0.5*UP+0.5*RIGHT).rotate(np.pi/6)
+ self.play(ShowCreation(angle),ShowCreation(theta))
+ self.wait(1)
+ line1 = Line().scale(1.25).rotate(np.pi/(2)).shift(1.8*UP+1.47*RIGHT)
+ line1.rotate(np.pi/8)
+ self.play(ShowCreation(line1))
+ self.wait(1.3)
+ v3 = self.add_vector(VECTORS[2],color = BLUE, stroke_width = 3.5)
+ text1 = TextMobject(r"$\vec{v}$cos$\theta$",color=BLUE).scale(0.55).shift(1.25*RIGHT+0.26*UP).rotate(np.pi/9)
+ self.play(ShowCreation(text1))
+ self.wait(2)
+ text2 = TextMobject(r"\text{Projection on }",r"\text{$\vec{v}$}",r"\text{ onto }",r"\text{$\vec{u}$}",r"\text{ = }",r"\text{$ |\vec{v}|cos\theta$}").scale(0.6).shift(4*RIGHT+3*UP)
+ text2[1].set_color(RED)
+ text2[3].set_color(YELLOW)
+ text2[5].set_color(BLUE)
+ text2.add_background_rectangle()
+ self.play(ShowCreation(text2))
+ self.wait(4)
+
+
+
\ No newline at end of file diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Conjugate_Symmetry_and_Positivity_of_Inner_Product.mp4 b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Conjugate_Symmetry_and_Positivity_of_Inner_Product.mp4 Binary files differnew file mode 100644 index 0000000..6b44c55 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Conjugate_Symmetry_and_Positivity_of_Inner_Product.mp4 diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/InnerProduct.gif b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/InnerProduct.gif Binary files differnew file mode 100644 index 0000000..55b6546 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/InnerProduct.gif diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Inner_Product_Space_Example.mp4 b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Inner_Product_Space_Example.mp4 Binary files differnew file mode 100644 index 0000000..edecd88 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Inner_Product_Space_Example.mp4 diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Instances_of_Topological_Spaces.gif b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Instances_of_Topological_Spaces.gif Binary files differnew file mode 100644 index 0000000..82d5c75 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Instances_of_Topological_Spaces.gif diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Interpretation_of_norm_as_length.gif b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Interpretation_of_norm_as_length.gif Binary files differnew file mode 100644 index 0000000..9eda7c5 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Interpretation_of_norm_as_length.gif diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Linerity_of_Inner_Product.mp4 b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Linerity_of_Inner_Product.mp4 Binary files differnew file mode 100644 index 0000000..a1e7b29 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Linerity_of_Inner_Product.mp4 diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Projection.gif b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Projection.gif Binary files differnew file mode 100644 index 0000000..c32fd6b --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Projection.gif diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Projection_of_vectors_in-3D_plane.mp4 b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Projection_of_vectors_in-3D_plane.mp4 Binary files differnew file mode 100644 index 0000000..bdca5f9 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/gifs2/Projection_of_vectors_in-3D_plane.mp4 diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/MCQ/Basis of a Vector Space and its Subspace.docx b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/MCQ/Basis of a Vector Space and its Subspace.docx Binary files differnew file mode 100644 index 0000000..0dbfc4f --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/MCQ/Basis of a Vector Space and its Subspace.docx diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/MCQ/Dual of a Space MCQ Questions.docx b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/MCQ/Dual of a Space MCQ Questions.docx Binary files differnew file mode 100644 index 0000000..1731a8f --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/MCQ/Dual of a Space MCQ Questions.docx diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/MCQ/Inner Product Space MCQ Questions.docx b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/MCQ/Inner Product Space MCQ Questions.docx Binary files differnew file mode 100644 index 0000000..80ee81c --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/MCQ/Inner Product Space MCQ Questions.docx diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/MCQ/Polynomial and Function Spaces MCQ Questions.docx b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/MCQ/Polynomial and Function Spaces MCQ Questions.docx Binary files differnew file mode 100644 index 0000000..7904fd1 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/MCQ/Polynomial and Function Spaces MCQ Questions.docx diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/MCQ/Vector Spaces MCQ Questions.docx b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/MCQ/Vector Spaces MCQ Questions.docx Binary files differnew file mode 100644 index 0000000..338b6f3 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/MCQ/Vector Spaces MCQ Questions.docx diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/Basis.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/Basis.py new file mode 100644 index 0000000..6d2edc8 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/Basis.py @@ -0,0 +1,174 @@ +from manimlib.imports import *
+
+
+VECTORS = [[1, 2],
+ [-4, 2],
+ [-3, -3],
+ [3,-2],
+ [3,3],
+ [-2,1],
+ [-1,-3]]
+
+class Scene1(LinearTransformationScene):
+
+ CONFIG = {
+ "include_background_plane": True,
+ "include_foreground_plane": False,
+ "show_coordinates": False,
+ "show_basis_vectors": True,
+ "basis_vector_stroke_width": 3,
+
+ }
+ def construct(self):
+ self.setup()
+ ihat, jhat = self.get_basis_vectors()
+ labels = self.get_basis_vector_labels()
+ self.add(ihat, jhat)
+ self.add(*labels)
+
+ self.show_vector_as_basis_sum()
+ self.wait(2)
+
+ def show_vector_as_basis_sum(self):
+ text1 = TextMobject(r"Vector Space $\mathbb{R}^2$}").scale(0.8).shift(3*UP)
+ text1.add_background_rectangle()
+ self.play(ShowCreation(text1))
+ text2 = TextMobject(r"$\mathbb{R}^2$",color=BLUE_E).scale(0.8).shift(6.5*LEFT+3.5*UP)
+ text3 = TextMobject(r"\text{Basis Vectors:}",r"\text{$\hat{i}$}",r"\text{,}",r"\text{$\hat{j}$}").scale(0.7).shift(2*UP+2.5*RIGHT)
+ text3[1].set_color(GREEN_C)
+ text3[3].set_color(RED_C)
+
+ self.wait(2)
+ self.play(Transform(text1,text2))
+ self.wait(1)
+ self.play(ShowCreation(text3))
+ self.wait(1.7)
+ self.play(FadeOut(text3))
+
+ for i in range(len(VECTORS)):
+ v = self.add_vector(VECTORS[i], stroke_width = 3,color=YELLOW_D)
+
+ linei = Line(start = ORIGIN, end = VECTORS[i][0]*RIGHT)
+ linei.set_color(GREEN_C)
+ linej = Line(start = linei.get_end(),
+ end = linei.get_end() + VECTORS[i][1]*UP)
+ linej.set_color(RED_C)
+ self.play(ShowCreation(linei))
+ self.play(ShowCreation(linej))
+ vlabel = self.get_vector_label(v, str(VECTORS[i][0]) +
+ r"\imath" + "+" +
+ str(VECTORS[i][1]) +
+ r"\jmath", at_tip = True)
+ self.play(ShowCreation(vlabel))
+
+ self.play(FadeOut(linei),FadeOut(linej))
+ self.wait(1)
+ dot = Dot(v.get_end(), fill_color = v.get_stroke_color())
+ self.play(ShowCreation(dot),FadeOut(v),FadeOut(vlabel))
+ self.wait(0.3)
+
+class Scene2(LinearTransformationScene):
+ CONFIG = {
+ "num_vectors" : 16,
+ "start_color" : GREY,
+ "end_color" : YELLOW_D,
+ "include_background_plane": True,
+ "include_foreground_plane": False,
+ }
+
+ def get_vectors(self):
+ return [
+ Vector([x, y], stroke_width = 3.5)
+ for x in np.arange(-int(FRAME_X_RADIUS), int(FRAME_X_RADIUS)+0.5, 0.5)
+ for y in np.arange(-int(FRAME_Y_RADIUS), int(FRAME_Y_RADIUS)+0.5, 0.5)
+ ]
+
+ def lock_in_faded_grid(self, dimness=0.7, axes_dimness=0.5):
+ plane = self.add_plane()
+ axes = plane.get_axes()
+ plane.fade(dimness)
+ axes.set_color(WHITE)
+ axes.fade(axes_dimness)
+ self.add(axes)
+
+ def construct(self):
+ self.lock_in_faded_grid()
+
+ vectors = self.get_vectors()
+ colors = Color(self.start_color).range_to(
+ self.end_color, len(vectors)
+ )
+ for vect, color in zip(vectors, colors):
+ vect.set_color(color)
+
+ vector_group = VGroup(*vectors)
+ self.play(
+ ShowCreation(
+ vector_group,
+ run_time = 3
+ )
+ )
+
+ self.wait(1)
+
+ vectors.sort(key=lambda v: v.get_length())
+ def v_to_dot(vector):
+ return Dot(vector.get_end(), fill_color = vector.get_stroke_color())
+ self.wait()
+ rate_functions = [
+ squish_rate_func(smooth, float(x)/(len(vectors)+2), 1)
+ for x in range(len(vectors))
+ ]
+ self.play(*[
+ Transform(v, v_to_dot(v), rate_func = rf, run_time = 3)
+ for v, rf in zip(vectors, rate_functions)
+ ])
+ self.wait(2)
+ self.play_final_animation(vectors, rate_functions)
+ self.wait(2)
+
+ text1 = TextMobject(" Basis is the minimum information required to ").shift(3.1*UP).scale(0.8)
+ text2 = TextMobject("generate the whole space.").scale(0.8).shift(2.6*UP)
+
+ text1.add_background_rectangle()
+ text2.add_background_rectangle()
+
+
+
+ self.play(ShowCreation(text1),ShowCreation(text2))
+
+ self.play(ShowCreation(self.get_basis_vectors()))
+ self.wait(3)
+
+ def play_final_animation(self, vectors, rate_functions):
+
+ h_line = Line(
+ FRAME_X_RADIUS*RIGHT, FRAME_X_RADIUS*LEFT,
+ stroke_width = 0.5,
+ color = BLUE_E
+ )
+ v_line = Line(
+ FRAME_Y_RADIUS*UP, FRAME_Y_RADIUS*DOWN,
+ stroke_width = 0.5,
+ color = BLUE_E
+ )
+ line_pairs = [
+ VGroup(h_line.copy().shift(y), v_line.copy().shift(x))
+ for v in vectors
+ for x, y, z in [v.get_center()]
+ ]
+ plane = NumberPlane()
+
+ self.play(
+ ShowCreation(plane),
+ *[
+ Transform(v, p, rate_func = rf)
+ for v, p, rf in zip(vectors, line_pairs, rate_functions)
+ ]
+ )
+ self.remove(*vectors)
+
+
+
+
+
\ No newline at end of file diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/Intersection_of_Subspaces.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/Intersection_of_Subspaces.py new file mode 100644 index 0000000..ec82daa --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/Intersection_of_Subspaces.py @@ -0,0 +1,52 @@ +from manimlib.imports import *
+class ThreeDSpace(ThreeDScene):
+
+
+ def construct(self):
+ axes = ThreeDAxes()
+ self.set_camera_orientation(phi = 80*DEGREES,theta =110*DEGREES)
+ self.begin_ambient_camera_rotation(rate=0.09)
+
+
+
+
+ cube = Cube(stroke_width=5,color=WHITE).shift([1.5,1.5,1.5]).scale(1.5)
+ cube.set_fill(TEAL)
+
+ cube.set_opacity(0.4)
+
+
+ plane1 = Polygon([0,0,3],[3,0,3],[3,3,0],[0,3,0])
+ plane1.set_opacity(0.65)
+ plane1.set_fill(PURPLE)
+ plane1.set_color(PURPLE)
+
+
+ plane2 = Polygon([0,3,3],[3,3,3],[3,0,0],[0,0,0])
+ plane2.set_opacity(0.7)
+ plane2.set_fill(RED)
+ plane2.set_color(RED)
+ line = Line(color=YELLOW,set_opacity=100).shift([1.5,1.5,1.5]).scale(1.5)
+
+ vgroup = VGroup(plane1,plane2,line,cube)
+ vgroup.shift([-1,-1,-1])
+
+ dot = Dot(color=BLACK).shift([0.5,0.5,0.5]).scale(1)
+ text = TextMobject(r"\text{The}",r"\text{line}",r"\text{representing the intersection of the two planes is a Subspace.}",opacity = 0.6).scale(0.7).shift(3*UP)
+ text[1].set_color(YELLOW)
+ self.add_fixed_in_frame_mobjects(text)
+ self.play(ShowCreation(text))
+
+
+ self.play(ShowCreation(cube))
+ self.play(ShowCreation(plane1))
+ self.play(ShowCreation(plane2))
+
+ self.play(ShowCreation(line),ShowCreation(dot))
+
+
+ self.wait(15)
+
+
+
+
\ No newline at end of file diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/Straight_Line_through_Origin.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/Straight_Line_through_Origin.py new file mode 100644 index 0000000..5790d2e --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/Straight_Line_through_Origin.py @@ -0,0 +1,48 @@ +from manimlib.imports import *
+from scipy import exp, sin, log,tan,cos
+class Straight_Line(GraphScene):
+ CONFIG = {
+ "x_min" : -4,
+ "x_max" : 4,
+ "y_min" : -4,
+ "y_max" : 4,
+ "y_tick_frequency" : 1,
+ "x_tick_frequency" : 1,
+ "x_labeled_nums" : list(np.arange(-4,5,1)),
+ "y_labeled_nums" : list(np.arange(-4,5,1)),
+ "graph_origin" : ORIGIN+0.7*DOWN,
+ "axes_color" : GREY,
+ "x_axis_width": 6,
+ "y_axis_height":6,
+ }
+ def construct(self):
+ self.setup_axes(animate=True)
+ line_1 = self.get_graph(lambda x : x, x_min=-3,x_max=3,color=YELLOW)
+ self.play(ShowCreation(line_1))
+ text1 = TextMobject("ax + by = 0",color=BLUE_B)
+ text1.shift(3*RIGHT+2*UP)
+ text1.scale(0.65)
+ dot = Dot(color=BLUE_B).shift(0.7*DOWN)
+ dot.scale(1.3)
+ self.play(ShowCreation(dot))
+ text2 = TextMobject("Line passing through the origin")
+ text2.scale(0.7)
+ text2.shift(3.5*UP)
+ self.play(ShowCreation(text1),ShowCreation(text2))
+ self.wait(1)
+ self.play(FadeOut(line_1),FadeOut(text2),FadeOut(text1))
+ text4=TextMobject("Line not passing through the origin")
+ text4.scale(0.7)
+ text4.shift(3.5*UP)
+ self.play(ShowCreation(text4))
+
+ line_2 = self.get_graph(lambda x : 2.5*x +1, x_min = -2, x_max=1, color = RED)
+ text3 = TextMobject(r"ax + by $\neq 0$",color=BLUE_B)
+ text3.scale(0.65)
+ self.play(ShowCreation(line_2))
+ text3.shift(1.5*RIGHT+2.2*UP)
+ self.play(ShowCreation(text3))
+ self.wait(1)
+
+
+
\ No newline at end of file diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/Subspace_Non_Example.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/Subspace_Non_Example.py new file mode 100644 index 0000000..115a722 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/Subspace_Non_Example.py @@ -0,0 +1,25 @@ +from manimlib.imports import *
+class ThreeDSpace(ThreeDScene):
+
+
+ def construct(self):
+ axes = ThreeDAxes()
+ self.add(axes)
+ self.set_camera_orientation(phi = 80*DEGREES,theta =110*DEGREES)
+ self.begin_ambient_camera_rotation(rate=0.09)
+ plane1 = Polygon([-1.5,1.5,-1.5],[1.5,1.5,-1.5],[1.5,-1.5,1.5],[-1.5,-1.5,1.5])
+ plane1.set_opacity(0.65)
+ plane1.set_fill(GREEN)
+ plane1.set_color(GREEN)
+
+
+ plane2 = Polygon([-1.5,1.5,1.5],[1.5,1.5,1.5],[1.5,-1.5,-1.5],[-1.5,-1.5,-1.5])
+ plane2.set_opacity(0.7)
+ plane2.set_fill(MAROON_A)
+ plane2.set_color(MAROON_A)
+ line = Line(color=YELLOW,set_opacity=100,start=[1.5,1.2,1.2],end=[-1.5,1.2,1.2])
+
+
+ self.play(ShowCreation(plane1),ShowCreation(plane2),ShowCreation(line))
+ self.wait(10)
+
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/Unit_Circle.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/Unit_Circle.py new file mode 100644 index 0000000..2973f08 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/Unit_Circle.py @@ -0,0 +1,68 @@ +from manimlib.imports import *
+import numpy as np
+import math
+
+class Unit_Circle(GraphScene):
+ CONFIG = {
+ "x_min" : -3,
+ "x_max" : 3,
+ "y_min" : -3,
+ "y_max" : 3,
+ "y_tick_frequency" : 1,
+ "x_tick_frequency" : 1,
+ "x_labeled_nums" : list(np.arange(-3,4,1)),
+ "y_labeled_nums" : list(np.arange(-3,4,1)),
+ "graph_origin" : ORIGIN,
+ "axes_color" : GREY,
+ "x_axis_width": 6,
+ "y_axis_height":6,
+ }
+
+ def construct(self):
+ self.setup_axes(animate = True)
+ circle = Circle(radius=1,color=BLUE)
+ self.play(ShowCreation(circle))
+ dot1 = Dot(color=RED).scale(0.7)
+ dot1.shift(1*UP)
+ dot2 = Dot(color=RED).scale(0.7)
+ dot2.shift(1*LEFT)
+ dot3 = Dot(color=RED).scale(0.7)
+ dot3.shift(1*DOWN)
+ dot4 = Dot(color=RED).scale(0.7)
+ dot4.shift(1*RIGHT)
+ dot5= Dot(color=RED).scale(0.7)
+ dot6 = Dot(color=RED).scale(0.7)
+ dot5.shift(0.5*RIGHT+(math.sqrt(3)/2)*UP)
+ dot6.shift(0.5*LEFT+(math.sqrt(3)/2)*DOWN)
+ dot7 = Dot(color=RED).scale(0.7)
+ dot7.shift(math.sqrt(2)/2*RIGHT+math.sqrt(2)/2*UP)
+ dot8 = Dot(color=RED).scale(0.7)
+ dot8.shift(math.sqrt(2)/2*LEFT+math.sqrt(2)/2*UP)
+ dot9 = Dot(color=RED).scale(0.7)
+ dot9.shift(0.5*LEFT+(math.sqrt(3)/2)*UP)
+ dot10 = Dot(color=RED).scale(0.7)
+ dot10.shift(math.sqrt(3)/2*LEFT+0.5*UP)
+ dot11=Dot(color=RED).scale(0.7)
+ dot11.shift(math.sqrt(3)/2*RIGHT+0.5*UP)
+ dot12= Dot(color=RED).scale(0.7)
+ dot12.shift(math.sqrt(3)/2*LEFT+0.5*DOWN)
+ dot13=Dot(color=RED).scale(0.7)
+ dot13.shift(math.sqrt(2)/2*RIGHT+math.sqrt(2)/2*DOWN)
+ dot14=Dot(color=RED).scale(0.7)
+ dot14.shift(math.sqrt(2)/2*LEFT+math.sqrt(2)/2*DOWN)
+ dot15=Dot(color=RED).scale(0.7)
+ dot15.shift(math.sqrt(3)/2*RIGHT+0.5*DOWN)
+ dot16=Dot(color=RED).scale(0.7)
+ dot16.shift(0.5*RIGHT+(math.sqrt(3)/2)*DOWN)
+ self.play(ShowCreation(dot1),ShowCreation(dot2))
+ self.play(ShowCreation(dot3),ShowCreation(dot4))
+ self.play(ShowCreation(dot5),ShowCreation(dot6))
+ self.play(ShowCreation(dot7),ShowCreation(dot8))
+ self.play(ShowCreation(dot9),ShowCreation(dot10))
+ self.play(ShowCreation(dot11),ShowCreation(dot12))
+ self.play(ShowCreation(dot13),ShowCreation(dot14))
+ self.play(ShowCreation(dot15),ShowCreation(dot16))
+ self.wait(4)
+
+
+
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Basis_generating_the_whole_2D_space.mp4 b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Basis_generating_the_whole_2D_space.mp4 Binary files differnew file mode 100644 index 0000000..9a43846 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Basis_generating_the_whole_2D_space.mp4 diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Intersection_of_Subspaces.mp4 b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Intersection_of_Subspaces.mp4 Binary files differnew file mode 100644 index 0000000..d43bfbe --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Intersection_of_Subspaces.mp4 diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Linear-Dependence-and-Independence.mp4 b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Linear-Dependence-and-Independence.mp4 Binary files differnew file mode 100644 index 0000000..c44801f --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Linear-Dependence-and-Independence.mp4 diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Straight_Line_Through_Origin.gif b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Straight_Line_Through_Origin.gif Binary files differnew file mode 100644 index 0000000..b7695a4 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Straight_Line_Through_Origin.gif diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Subspace_Non_Example.mp4 b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Subspace_Non_Example.mp4 Binary files differnew file mode 100644 index 0000000..390f4cf --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Subspace_Non_Example.mp4 diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Unit_Circle.gif b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Unit_Circle.gif Binary files differnew file mode 100644 index 0000000..165d040 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/gifs/Unit_Circle.gif diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/linear_dependence_and_independence.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/linear_dependence_and_independence.py new file mode 100644 index 0000000..b092777 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Subspaces/linear_dependence_and_independence.py @@ -0,0 +1,293 @@ +from manimlib.imports import *
+class Linear_Dependence(ThreeDScene):
+
+ def construct(self):
+
+ axes = ThreeDAxes()
+ axes.set_stroke(width=1,color=GOLD)
+ self.add(axes)
+ self.set_camera_orientation(phi = 70*DEGREES,theta =110*DEGREES)
+ self.begin_ambient_camera_rotation(rate=0.05)
+ self.wait(0.5)
+ text1 = TextMobject(r"Consider 3 linearly dependent vectors in $\mathbb{R}^3$").scale(0.5).shift(2.5*UP+4*LEFT)
+ line1 = Line(color = YELLOW,opacity=350,start = ORIGIN,end = [-1,1,1])
+ self.add_fixed_in_frame_mobjects(text1)
+ self.play(ShowCreation(text1))
+ self.play(ShowCreation(line1))
+ self.wait(1)
+ line2 = Line(color = RED,opacity=350,start = ORIGIN,end = [1,0.5,0.5])
+ self.play(ShowCreation(line2))
+ self.wait(1)
+ line3 = Line(color = BLUE,opacity=350,start = ORIGIN,end = [0.5,1,1])
+ self.play(ShowCreation(line3))
+ self.wait(1)
+ text2 = TextMobject("Scaling the Vectors").scale(0.5).shift(2.5*UP+4.5*LEFT)
+ text3 = TextMobject(r"\text{and}",r"\text{Adding}",r"\text{them}").scale(0.5).shift(2*UP+4.5*LEFT)
+ text3[1].set_color(GREEN)
+
+
+ line4 = Line(color = YELLOW,opacity=350,start = ORIGIN,end = [-2,2,2])
+ line5 = Line(color = RED,opacity = 350,start = ORIGIN,end = [3,1.5,1.5])
+ line6 = Line(color = BLUE,opacity = 350,start = ORIGIN,end = [-0.5,-1,-1])
+ self.wait(1.1)
+ self.play(FadeOut(text1))
+ self.add_fixed_in_frame_mobjects(text2)
+ self.play(ShowCreation(text2))
+ self.wait(0.7)
+ self.play(Transform(line1,line4))
+ self.wait(0.5)
+ self.play(Transform(line2,line5))
+ self.wait(0.5)
+ self.play(Transform(line3,line6))
+
+ self.wait(1.5)
+
+ line7 = Line(color = RED,opacity = 350,start = [-2,2,2],end = [1,3.5,3.5])
+ line8 = Line(color = BLUE,opacity = 350,start = [1,3.5,3.5],end = [0.5,2.5,2.5])
+ line9 = Line(color = GREEN,opacity = 350,start = ORIGIN,end = [0.5,2.5,2.5])
+
+ self.add_fixed_in_frame_mobjects(text3)
+ self.play(ShowCreation(text3))
+ self.wait(0.7)
+ self.play(FadeOut(line2),ShowCreation(line7))
+ self.play(FadeOut(line3),ShowCreation(line8))
+ self.wait(0.7)
+ self.play(ShowCreation(line9))
+ self.wait(1.2)
+
+
+ plane = Polygon([-4,4,4],[4,4,4],[4,-4,-4],[-4,-4,-4])
+ plane.set_opacity(0.4)
+ plane.set_fill(DARK_BLUE)
+ plane.set_color(DARK_BLUE)
+ self.play(ShowCreation(plane))
+ self.wait(1.2)
+ text4 = TextMobject("Linearly Combinations").scale(0.65).shift(3.5*UP+4*RIGHT)
+
+ text5 = TextMobject(r"\text{$\vec{z_1}$}",r"\text{=}",r"\text{2}",r"\text{$\vec{u}$}",r"\text{+}",r"\text{3}",r"\text{$\vec{v}$}",r"\text{+}",r"\text{(-1)}",r"\text{$\vec{w}$}",).scale(0.55).shift(2.5*UP+3.9*RIGHT)
+ text5[0].set_color(GREEN)
+ text5[3].set_color(YELLOW)
+ text5[6].set_color(RED)
+ text5[9].set_color(BLUE)
+ self.add_fixed_in_frame_mobjects(text5)
+ self.add_fixed_in_frame_mobjects(text4)
+ self.play(ShowCreation(text4),ShowCreation(text5))
+ self.wait(3.8)
+ bunch1 = VGroup(line7,line8,line9)
+ line10 = Line(color=YELLOW,opacity=350,start=ORIGIN,end=[-0.5,0.5,0.5])
+ line11 = Line(color=RED,opacity=350,start=ORIGIN,end=[2,1,1])
+ line12 = Line(color = BLUE,opacity = 350,start = ORIGIN,end = [-1.5,-3,-3])
+ line13 = Line(color = RED,opacity = 350,start = [-0.5,0.5,0.5],end = [1.5,1.5,1.5])
+ line14 = Line(color = BLUE,opacity = 350,start = [1.5,1.5,1.5],end = [0,-1.5,-1.5])
+ line15 = Line(color = GREEN,opacity = 350,start = ORIGIN,end = [0,-1.5,-1.5])
+ bunch2 = VGroup(line13,line14,line15,line10)
+ self.play(FadeOut(line1),Transform(bunch1,bunch2))
+ self.wait(1.2)
+ text6 = TextMobject(r"\text{$\vec{z_2}$}",r"\text{=}",r"\text{0.5}",r"\text{$\vec{u}$}",r"\text{+}",r"\text{2}",r"\text{$\vec{v}$}",r"\text{+}",r"\text{(-3)}",r"\text{$\vec{w}$}",).scale(0.55).shift(2*UP+4*RIGHT)
+ text6[0].set_color(GREEN)
+ text6[3].set_color(YELLOW)
+ text6[6].set_color(RED)
+ text6[9].set_color(BLUE)
+
+ self.add_fixed_in_frame_mobjects(text6)
+ self.play(ShowCreation(text6))
+ self.wait(2.8)
+ bunch3 = VGroup(line10,bunch1)
+
+ line16 = Line(color=YELLOW,opacity=350,start=ORIGIN,end=[-2,2,2])
+ line17 = Line(color=RED,opacity=350,start=ORIGIN,end=[1,0.5,0.5])
+
+ line18 = Line(color = RED,opacity = 350,start = [-2,2,2],end = [-1,2.5,2.5])
+
+ line19 = Line(color = GREEN,opacity = 350,start = ORIGIN,end = [-1,2.5,2.5])
+ bunch4 = VGroup(line16,line18,line19)
+ self.play(Transform(bunch3,bunch4))
+ self.wait(1.2)
+ text7 = TextMobject(r"\text{$\vec{z_3}$}",r"\text{=}",r"\text{(-2)}",r"\text{$\vec{u}$}",r"\text{+}",r"\text{2}",r"\text{$\vec{v}$}",r"\text{+}",r"\text{(0)}",r"\text{$\vec{w}$}",).scale(0.55).shift(1.5*UP+4*RIGHT)
+ text7[0].set_color(GREEN)
+ text7[3].set_color(YELLOW)
+ text7[6].set_color(RED)
+ text7[9].set_color(BLUE)
+ self.add_fixed_in_frame_mobjects(text7)
+ self.play(ShowCreation(text7))
+
+ self.wait(5)
+ self.play(FadeOut(text2),FadeOut(text3))
+ text8 = TextMobject(r"\text{$\vec{z}$}",r"\text{=}",r"\text{a}",r"\text{$\vec{u}$}",r"\text{+}",r"\text{b}",r"\text{$\vec{v}$}",r"\text{+}",r"\text{c}",r"\text{$\vec{w}$}",).scale(0.55).shift(0.8*UP+4*RIGHT)
+ text8[0].set_color(GREEN)
+ text8[3].set_color(YELLOW)
+ text8[6].set_color(RED)
+ text8[9].set_color(BLUE)
+ rect1 = Rectangle(height=0.7,stroke_width=1.2)
+ rect1.surround(text8)
+ self.add_fixed_in_frame_mobjects(rect1)
+ self.add_fixed_in_frame_mobjects(text8)
+ self.play(ShowCreation(text8))
+ text9 = TextMobject(r"\text{All the}",r"\text{linear combinations}",r"\text{of linearly dependent vectors}",r"\text{$\vec{u},$}",r"\text{$\vec{v},$}",r"\text{and}",r"\text{$\vec{w}$}",r"\text{spans}").scale(0.5).shift(2.5*DOWN)
+ text9[1].set_color(GREEN)
+ text9[3].set_color(YELLOW)
+ text9[4].set_color(RED)
+ text9[6].set_color(BLUE)
+ text9.add_background_rectangle()
+ text10 = TextMobject(r"this entire plane (and not the entire $\mathbb{R}^3 )$.").scale(0.5).shift(3*DOWN)
+ text10.add_background_rectangle()
+ self.wait(2)
+ self.add_fixed_in_frame_mobjects(text10)
+
+ self.add_fixed_in_frame_mobjects(text9)
+ self.play(ShowCreation(text9),ShowCreation(text10))
+ self.wait(7)
+
+class Linear_Independence(ThreeDScene):
+
+ def construct(self):
+
+ axes = ThreeDAxes()
+ axes.set_stroke(width=1,color=GOLD)
+ self.add(axes)
+ self.set_camera_orientation(phi = 70*DEGREES,theta =110*DEGREES)
+ self.begin_ambient_camera_rotation(rate=0.1)
+ self.wait(0.5)
+ text1 = TextMobject(r"Consider 3 linearly independent vectors in $\mathbb{R}^3$").scale(0.5).shift(2.5*UP+4*LEFT)
+ self.add_fixed_in_frame_mobjects(text1)
+ self.play(ShowCreation(text1))
+ line1 = Line(color = YELLOW,opacity=350,start = ORIGIN,end = [-1,1,1])
+ self.play(ShowCreation(line1))
+ self.wait(0.7)
+ line2 = Line(color = RED,opacity=350,start = ORIGIN,end = [1,0.5,0.5])
+ self.play(ShowCreation(line2))
+ self.wait(0.7)
+ line3 = Line(color = BLUE,opacity=350,start = ORIGIN,end = [0,-1,-2])
+ self.play(ShowCreation(line3))
+ self.wait(0.7)
+
+ self.wait(1)
+ self.play(FadeOut(text1))
+ text2 = TextMobject("Scaling the Vectors").scale(0.5).shift(2.5*UP+5*LEFT)
+ self.add_fixed_in_frame_mobjects(text2)
+ self.play(ShowCreation(text2))
+ self.wait(0.7)
+ line4 = Line(color = YELLOW,opacity=350,start = ORIGIN,end = [-0.5,0.5,0.5])
+ self.play(Transform(line1,line4))
+ self.wait(0.7)
+ line5 = Line(color = RED,opacity=350,start = ORIGIN,end = [2,1,1])
+ self.play(Transform(line2,line5))
+ self.wait(0.7)
+ line6 = Line(color = BLUE,opacity=350,start = ORIGIN,end = [0,-1.5,-3])
+ self.play(Transform(line3,line6))
+ self.wait(1.7)
+ text3 = TextMobject(r"\text{and}",r"\text{Adding}",r"\text{them}").scale(0.5).shift(2*UP+5*LEFT)
+ text3[1].set_color(GREEN)
+
+ line7 = Line(color = RED,opacity = 350,start = [-0.5,0.5,0.5],end = [1.5,1.5,1.5])
+ line8 = Line(color = BLUE,opacity = 350,start = [1.5,1.5,1.5],end = [1.5,0,-1.5])
+ line9 = Line(color = GREEN,opacity = 350,start = ORIGIN,end = [1.5,0,-1.5])
+ self.add_fixed_in_frame_mobjects(text3)
+ self.play(ShowCreation(text3))
+ self.wait(0.7)
+ self.play(FadeOut(line2),ShowCreation(line7))
+ self.play(FadeOut(line3),ShowCreation(line8))
+ self.wait(0.7)
+ self.play(ShowCreation(line9))
+ self.wait(1.5)
+ plane = Polygon([-4,4,4],[4,4,4],[4,-4,-4],[-4,-4,-4])
+ plane.set_opacity(0.3)
+ plane.set_fill(DARK_BLUE)
+ plane.set_color(DARK_BLUE)
+ self.play(ShowCreation(plane))
+ self.wait(1.2)
+ text4 = TextMobject("Linearly Combinations").scale(0.65).shift(3.5*UP+4*RIGHT)
+ self.add_fixed_in_frame_mobjects(text4)
+ self.play(ShowCreation(text4))
+ self.wait(0.7)
+ text5 = TextMobject(r"\text{$\vec{z_1}$}",r"\text{=}",r"\text{0.5}",r"\text{$\vec{u}$}",r"\text{+}",r"\text{2}",r"\text{$\vec{v}$}",r"\text{+}",r"\text{1.5}",r"\text{$\vec{w}$}").scale(0.55).shift(2.5*UP+3.8*RIGHT)
+ text5[0].set_color(GREEN)
+ text5[3].set_color(YELLOW)
+ text5[6].set_color(RED)
+ text5[9].set_color(BLUE)
+ self.add_fixed_in_frame_mobjects(text5)
+
+ self.play(ShowCreation(text5))
+ self.wait(3.8)
+ bunch1 = VGroup(line7,line8,line9)
+ line10 = Line(color=YELLOW,opacity=350,start=ORIGIN,end=[-3,3,3])
+
+ line11 = Line(color = BLUE,opacity = 350,start = [-3,3,3],end = [-2,3.5,3.5])
+
+
+ line12 = Line(color = GREEN,opacity = 350,start = ORIGIN,end = [-2,3.5,3.5])
+ bunch2 = VGroup(line10,line11,line12)
+ self.play(FadeOut(line1),Transform(bunch1,bunch2))
+ self.wait(1.2)
+
+ text6 = TextMobject(r"\text{$\vec{z_2}$}",r"\text{=}",r"\text{3}",r"\text{$\vec{u}$}",r"\text{+}",r"\text{1}",r"\text{$\vec{v}$}",r"\text{+}",r"\text{0}",r"\text{$\vec{w}$}").scale(0.55).shift(2*UP+3.6*RIGHT)
+ text6[0].set_color(GREEN)
+ text6[3].set_color(YELLOW)
+ text6[6].set_color(RED)
+ text6[9].set_color(BLUE)
+
+ self.add_fixed_in_frame_mobjects(text6)
+ self.play(ShowCreation(text6))
+ self.wait(2.8)
+ bunch3 = VGroup(line10,bunch1)
+
+ line13 = Line(color=YELLOW,opacity=350,start=ORIGIN,end=[2.5,-2.5,-2.5])
+ line14 = Line(color=RED,opacity=350,start=[2.5,-2.5,-2.5],end=[0.5,-3.5,-3.5])
+
+ line15 = Line(color = BLUE,opacity = 350,start = [0.5,-3.5,-3.5],end = [0.5,-0.5,2.5])
+
+ line16 = Line(color = GREEN,opacity = 350,start = ORIGIN,end = [0.5,-0.5,2.5])
+ bunch4 = VGroup(line13,line14,line15,line16)
+ self.play(Transform(bunch3,bunch4))
+ self.wait(1.2)
+ text7 = TextMobject(r"\text{$\vec{z_3}$}",r"\text{=}",r"\text{(-2.5)}",r"\text{$\vec{u}$}",r"\text{+}",r"\text{(-2)}",r"\text{$\vec{v}$}",r"\text{+}",r"\text{(-3)}",r"\text{$\vec{w}$}").scale(0.55).shift(1.5*UP+4.2*RIGHT)
+ text7[0].set_color(GREEN)
+ text7[3].set_color(YELLOW)
+ text7[6].set_color(RED)
+ text7[9].set_color(BLUE)
+ self.add_fixed_in_frame_mobjects(text7)
+ self.play(ShowCreation(text7))
+ self.play(FadeOut(text3),FadeOut(text2))
+ self.wait(1.2)
+ text8 = TextMobject(r"\text{$\vec{z}$}",r"\text{=}",r"\text{a}",r"\text{$\vec{u}$}",r"\text{+}",r"\text{b}",r"\text{$\vec{v}$}",r"\text{+}",r"\text{c}",r"\text{$\vec{w}$}").scale(0.55).shift(0.8*UP+3.6*RIGHT)
+ text8[0].set_color(GREEN)
+ text8[3].set_color(YELLOW)
+ text8[6].set_color(RED)
+ text8[9].set_color(BLUE)
+ rect1 = Rectangle(height=0.7,stroke_width=1.2)
+ rect1.surround(text8)
+ self.add_fixed_in_frame_mobjects(rect1)
+ self.add_fixed_in_frame_mobjects(text8)
+ self.play(ShowCreation(text8))
+ self.wait(1.7)
+ self.play(FadeOut(plane))
+ self.wait(1.5)
+ text9 = TextMobject(r"\text{All the}",r"\text{linear combinations}",r"\text{of linearly independent vectors}",r"\text{$\vec{u},$}",r"\text{$\vec{v},$}",r"\text{and}",r"\text{$\vec{w}$}",r"\text{spans}").scale(0.5).shift(2.5*DOWN)
+ text9[1].set_color(GREEN)
+ text9[3].set_color(YELLOW)
+ text9[4].set_color(RED)
+ text9[6].set_color(BLUE)
+ text9.add_background_rectangle()
+ text10 = TextMobject("not only the plane but the entire 3-D space.").scale(0.5).shift(3*DOWN)
+ text10.add_background_rectangle()
+ self.add_fixed_in_frame_mobjects(text9)
+
+ self.add_fixed_in_frame_mobjects(text10)
+ self.play(ShowCreation(text9),ShowCreation(text10))
+ self.wait(2.3)
+
+ self.play(FadeOut(text5),FadeOut(text6),FadeOut(text7))
+ self.wait(0.6)
+ text9 = TextMobject(r"\text{$\vec{z}$}",r"\text{=}",r"\text{a}",r"\text{$\vec{u}$}",r"\text{+}",r"\text{b}",r"\text{$\vec{v}$}",r"\text{+}",r"\text{c}",r"\text{$\vec{w}$}",r"\text{= 0}").scale(0.55).shift(2.6*UP+3.9*RIGHT)
+ text9[0].set_color(GREEN)
+ text9[3].set_color(YELLOW)
+ text9[6].set_color(RED)
+ text9[9].set_color(BLUE)
+ text10 = TextMobject("only when a = b = c = 0.").scale(0.55).shift(2.2*UP+3.9*RIGHT)
+ self.play(FadeOut(text8),FadeOut(rect1))
+ self.wait(0.7)
+ self.add_fixed_in_frame_mobjects(text9)
+ self.add_fixed_in_frame_mobjects(text10)
+
+ self.play(ShowCreation(text9),ShowCreation(text10))
+ self.wait(5)
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/3D_Vector_Space.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/3D_Vector_Space.py new file mode 100644 index 0000000..70913eb --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/3D_Vector_Space.py @@ -0,0 +1,14 @@ +from manimlib.imports import *
+class ThreeDSpace(ThreeDScene):
+ def construct(self):
+ curve = ParametricFunction(
+ lambda x: np.array([
+ 0, -x , x]), color = YELLOW, t_min = -2, t_max = 2)
+ axes = ThreeDAxes()
+ axes.set_stroke(width=1,color=GOLD)
+ self.add(axes)
+ self.set_camera_orientation(phi = 70*DEGREES,theta =60*DEGREES)
+ self.begin_ambient_camera_rotation(rate=0.3)
+ self.play(ShowCreation(curve))
+ self.wait(6)
+
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/Vector_Addition_and_Scalar_Multiplication.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/Vector_Addition_and_Scalar_Multiplication.py new file mode 100644 index 0000000..12c9bce --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/Vector_Addition_and_Scalar_Multiplication.py @@ -0,0 +1,177 @@ +from manimlib.imports import *
+VECTORS = [[1, 2],
+ [1, 1],
+ [2, 2],
+ [3, 4]]
+class Scene1(LinearTransformationScene):
+ CONFIG = {
+ "include_background_plane": True,
+ "include_foreground_plane": False,
+ "show_coordinates": True,
+ "show_basis_vectors": False,
+ "basis_vector_stroke_width": 3,
+ }
+ def construct(self):
+ self.write_stuff()
+ v1 = self.add_vector(VECTORS[0],color = YELLOW, stroke_width = 3.5)
+ line1 = Line(start = ORIGIN, end = VECTORS[0][0]*RIGHT)
+ line1.set_color(RED_B)
+ line2 = Line(start = line1.get_end(), end = line1.get_end() + VECTORS[0][1]*UP)
+ line2.set_color(GREEN_D)
+ self.play(ShowCreation(line1))
+ self.play(ShowCreation(line2))
+ self.wait(0.5)
+ text1 = TextMobject("(1, 2)",color=YELLOW).scale(0.6).shift(2*UP + 1.5*RIGHT)
+ self.play(ShowCreation(text1))
+ text2 = TextMobject(r"$\vec{a}$",color = YELLOW).scale(0.8).shift(2*UP + 1.2*RIGHT)
+ text3 = TextMobject(r"\text{$\vec{a}$}",r"\text{=}",r"\text{(1, 2)}").scale(0.7).shift(5.1*LEFT + 2.5*UP)
+ text3[0].set_color(YELLOW)
+ text3[2].set_color(YELLOW)
+ self.wait(1)
+ self.play(Transform(text1,text2),FadeOut(line1),FadeOut(line2))
+ self.wait(0.5)
+ self.play(ShowCreation(text3))
+ self.wait(1)
+
+
+ v2 = self.add_vector(VECTORS[1],color = BLUE, stroke_width = 3.5)
+ line3 = Line(start = ORIGIN, end = VECTORS[1][0]*RIGHT)
+ line3.set_color(RED_B)
+ line4 = Line(start = line3.get_end(), end = line3.get_end() + VECTORS[1][1]*UP)
+ line4.set_color(GREEN_D)
+ self.play(ShowCreation(line3))
+ self.play(ShowCreation(line4))
+ self.wait(0.5)
+ text4 = TextMobject("(2, 2)",color=BLUE).scale(0.6).shift(1*UP + 1.5*RIGHT)
+ self.play(ShowCreation(text4))
+ text5 = TextMobject(r"$\vec{b}$",color = BLUE).scale(0.8).shift(1*UP + 1.2*RIGHT)
+ text6 = TextMobject(r"\text{$\vec{b}$}",r"\text{=}",r"\text{(1, 1)}").scale(0.7).shift(5.1*LEFT + 1.8*UP)
+ text6[0].set_color(BLUE)
+ text6[2].set_color(BLUE)
+ self.wait(1)
+ self.play(Transform(text4,text5),FadeOut(line3),FadeOut(line4))
+ self.wait(0.5)
+ self.play(ShowCreation(text6))
+ self.wait(2)
+
+
+
+ text7 = TextMobject(r"\text{Scaling}",r"\text{ $\vec{b}$ }",r"\text{by 2 units}",color = GOLD).scale(0.7).shift(2.8*UP+4*RIGHT)
+ text7[1].set_color(BLUE)
+ self.play(ShowCreation(text7))
+ self.play(FadeOut(text4))
+ v3 = self.add_vector(VECTORS[2],color = BLUE, stroke_width = 3.5)
+ line7 = Line(start = ORIGIN, end = VECTORS[2][0]*RIGHT)
+ line7.set_color(RED_B)
+ line8 = Line(start = line7.get_end(), end = line7.get_end() + VECTORS[2][1]*UP)
+ line8.set_color(GREEN_D)
+ self.play(FadeOut(v2))
+ self.play(ShowCreation(line7))
+ self.play(ShowCreation(line8))
+
+
+ self.wait(0.5)
+
+ text8 = TextMobject("(2, 2)",color=BLUE).scale(0.6).shift(2.5*RIGHT + 2*UP)
+ text9 = TextMobject(r"$2\vec{b}$",color = BLUE).scale(0.8).shift(2*UP + 2.3*RIGHT)
+ self.play(ShowCreation(text8))
+ self.wait(1)
+ self.play(Transform(text8,text9),FadeOut(line7),FadeOut(line8))
+ self.wait(1)
+ text10 = TextMobject(r"\text{$2\vec{b}$}",r"\text{=}",r"\text{(2, 2)}").scale(0.7).shift(5.1*LEFT + 1.8*UP)
+ text10[0].set_color(BLUE)
+ text10[2].set_color(BLUE)
+ self.play(Transform(text6,text10))
+ self.wait(1)
+ self.play(FadeOut(text7))
+ self.wait(1.5)
+
+
+
+ text11 = TextMobject(r"Addition of vectors",color = GOLD).scale(0.7).shift(2.8*UP+5*RIGHT)
+ self.play(ShowCreation(text11))
+ v1.move_to(3*UP+2.5*RIGHT)
+ self.play(ShowCreation(v1),FadeOut(text8),FadeOut(text1))
+ v4 = self.add_vector(VECTORS[3],color = ORANGE, stroke_width = 3.5)
+ line7 = Line(start = ORIGIN, end = VECTORS[3][0]*RIGHT)
+ line7.set_color(RED_B)
+ line8 = Line(start = line7.get_end(), end = line7.get_end() + VECTORS[3][1]*UP)
+ line8.set_color(GREEN_D)
+ self.play(ShowCreation(line7))
+ self.play(ShowCreation(line8))
+ text12 = TextMobject("(3, 4)",color=ORANGE).scale(0.6).shift(3.5*RIGHT + 3.8*UP)
+ text13 = TextMobject(r"$\vec{c}$",color = ORANGE).scale(0.8).shift(3.7*UP + 3.3*RIGHT)
+ self.play(ShowCreation(text12))
+ self.wait(1)
+ self.play(Transform(text12,text13),FadeOut(line7),FadeOut(line8))
+ self.wait(1)
+ add = TextMobject("+").scale(0.8).shift(6.2*LEFT + 1.8*UP)
+ line_1= Line().shift(1.5*UP+5.1*LEFT).scale(1.5)
+ line_2= Line().shift(0.8*UP+5.1*LEFT).scale(1.5)
+
+ text14=TextMobject(r"\text{$\vec{c}$}",r"\text{=}",r"\text{$\vec{a}$}",r"\text{+}",r"\text{$2\vec{b}$}",r"\text{=}",r"\text{(3, 4)}").scale(0.7).shift(1.2*UP+5.1*LEFT)
+ text14[0].set_color(ORANGE)
+ text14[2].set_color(YELLOW)
+ text14[4].set_color(BLUE)
+ text14[6].set_color(ORANGE)
+
+
+ self.play(ShowCreation(add),ShowCreation(line_1))
+ self.play(ShowCreation(text14),ShowCreation(line_2))
+ self.wait(3)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ def write_stuff(self):
+ self.text = []
+ text = self.text
+
+ text.append(TexMobject(r"\text{Consider the Vector Space }",
+ r"{\mathbb{R}^2}"))
+ self.add_title(text[0], 0.7, animate = True)
+ text.append(TexMobject(r"{\mathbb{R}^2}",
+ tex_to_color_map = {r"{\mathbb{R}^2}": BLUE_E}))
+ text[1].shift(6.5*LEFT+3.5*UP)
+ self.wait(0.5)
+ self.play(Transform(text[0], text[1]))
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/Vector_Space_As_Functions.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/Vector_Space_As_Functions.py new file mode 100644 index 0000000..4f5614d --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/Vector_Space_As_Functions.py @@ -0,0 +1,62 @@ +from manimlib.imports import *
+from scipy import sin,cos
+class FunctionalVectorSpace(GraphScene):
+ CONFIG = {
+ "x_min": -5,
+ "x_max": 5,
+ "y_min": -5,
+ "y_max": 5,
+ "graph_origin": ORIGIN,
+ }
+ def construct(self):
+ self.setup_axes(animate = True)
+ curve1 = self.get_graph(lambda x : sin(x), x_min=-5,x_max=5,color=YELLOW_E)
+ curve2 = self.get_graph(lambda x : cos(x), x_min=-5,x_max=5,color=RED)
+ self.play(ShowCreation(curve1))
+ fx=TextMobject(r"$f(x)$",color=YELLOW_E).scale(0.7)
+ fx.shift(5*LEFT+0.7*UP)
+ self.play(ShowCreation(fx))
+ self.play(ShowCreation(curve2))
+ gx=TextMobject(r"$g(x)$",color=RED).scale(0.7)
+ gx.shift(5*LEFT+0.2*UP)
+ self.play(ShowCreation(gx))
+ self.wait(2)
+ scaling=TextMobject("Scaling f(x) by 2 units",color=GOLD).scale(0.65)
+ scaling.shift(3*LEFT+2.4*UP)
+ curve3 = self.get_graph(lambda x : 2*sin(x), x_min=-5,x_max=5,color=BLUE)
+ fx2=TextMobject(r"$2f(x)$",color=BLUE).scale(0.7)
+ fx2.shift(5*LEFT+1*UP)
+ self.play(Transform(curve1,curve3),FadeOut(fx),ShowCreation(fx2),ShowCreation(scaling))
+ self.wait(3)
+ hx = TextMobject(r"$h(x)$",color=PURPLE).scale(0.7)
+ hx.shift(4.9*LEFT+1.5*UP)
+ curve4 = self.get_graph(lambda x : 2*sin(x) + cos(x), x_min=-5,x_max=5,color=PURPLE)
+ self.play(ShowCreation(curve4),ShowCreation(hx))
+ self.play(FadeOut(curve2),FadeOut(curve1),FadeOut(fx2),FadeOut(gx),FadeOut(scaling))
+ hxn=TextMobject(r"$h(x)$",color=PURPLE).scale(0.7)
+ hxn.shift(3*RIGHT+2.4*UP)
+ equal = TextMobject("=").scale(0.7)
+ equal.shift(3.5*RIGHT+2.4*UP)
+ fx2n=TextMobject(r"$2f(x)$",color=BLUE).scale(0.7)
+ fx2n.shift(4.2*RIGHT+2.4*UP)
+ add=TextMobject("+").scale(0.7)
+ add.shift(4.8*RIGHT+2.4*UP)
+ gxn=TextMobject(r"$g(x)$",color=RED).scale(0.7)
+ gxn.shift(5.3*RIGHT+2.4*UP)
+ vector_add=TextMobject("Vector Addition",color=GOLD).scale(0.65)
+ vector_add.shift(3*UP+3*RIGHT)
+ self.play(ShowCreation(hxn),ShowCreation(equal),ShowCreation(fx2n),ShowCreation(add),ShowCreation(gxn),ShowCreation(vector_add))
+ self.wait(2)
+
+
+
+
+
+
+
+
+
+
+
+
+
\ No newline at end of file diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/gifs/3D_Vector_Space.gif b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/gifs/3D_Vector_Space.gif Binary files differnew file mode 100644 index 0000000..137546a --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/gifs/3D_Vector_Space.gif diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/gifs/Function_Vector_Space_Example.gif b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/gifs/Function_Vector_Space_Example.gif Binary files differnew file mode 100644 index 0000000..d9edf46 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/gifs/Function_Vector_Space_Example.gif diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/gifs/Vector_Addition_and_Scalar_Multiplication.mp4 b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/gifs/Vector_Addition_and_Scalar_Multiplication.mp4 Binary files differnew file mode 100644 index 0000000..8acbb81 --- /dev/null +++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Vector-Spaces/gifs/Vector_Addition_and_Scalar_Multiplication.mp4 |
