diff options
Diffstat (limited to 'FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces')
10 files changed, 315 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 |