diff options
Diffstat (limited to 'FSF-2020')
8 files changed, 293 insertions, 0 deletions
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/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/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/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)
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