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Diffstat (limited to 'FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces')
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diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/README.md b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/README.md new file mode 100644 index 0000000..6c90bf4 --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/README.md @@ -0,0 +1,30 @@ +# Contributer: Archit Sangal +My Github Account : <a href="https://github.com/architsangal">architsangal</a> (https://github.com/architsangal) +<br/></br> + +## Sub-Topics Covered: ++ The Four Fundamental Subspaces + +#### Video 1: Writing System of linear equations in form of Ax=b + + +#### Video 2: Column Space is same as the image of Linear Transformation + + +#### Video 3: Existance of Solution w.r.t. vector b + + +#### Video 4: Null Space (Visually) + + +#### Video 5: Definition 1 is equivalent to Definition 2 + + +#### Video 6: Relation between Row Space and Null Space + + +#### Fig. 1 Naming of the left null space + + +#### Video 7: Left Null Space(Visually) +
\ No newline at end of file diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file10_Left_Null_Space_pSv8iio_d5Sy9qS.gif b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file10_Left_Null_Space_pSv8iio_d5Sy9qS.gif Binary files differnew file mode 100644 index 0000000..3f50635 --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file10_Left_Null_Space_pSv8iio_d5Sy9qS.gif diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file11.gif b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file11.gif Binary files differnew file mode 100644 index 0000000..8f74202 --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file11.gif diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file12.gif b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file12.gif Binary files differnew file mode 100644 index 0000000..aa7403b --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file12.gif diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file13.gif b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file13.gif Binary files differnew file mode 100644 index 0000000..34b54c7 --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file13.gif diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file14.gif b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file14.gif Binary files differnew file mode 100644 index 0000000..b77791b --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file14.gif diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file15.gif b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file15.gif Binary files differnew file mode 100644 index 0000000..8bb24bf --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file15.gif diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file16.gif b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file16.gif Binary files differnew file mode 100644 index 0000000..87e0026 --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file16.gif diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file17.gif b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file17.gif Binary files differnew file mode 100644 index 0000000..eec819a --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file17.gif diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file18_NOT_in_lecture_note_Column_Space.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file18_NOT_in_lecture_note_Column_Space.py new file mode 100644 index 0000000..afe4f9a --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file18_NOT_in_lecture_note_Column_Space.py @@ -0,0 +1,30 @@ +from manimlib.imports import * + +class Column_Space(Scene): + def construct(self): + + A = TextMobject(r"$A = $",r"$\left( \begin{array}{c c c} 1 & 2 & 1 \\ 1 & 3 & 1 \\ 2 & 1 & 4 \\ 3 & 2 & 3 \end{array} \right)$") + A.move_to(2*UP) + A[1].set_color(color = DARK_BLUE) + A.scale(0.75) + + self.play(Write(A),run_time = 1) + + CS_A = TextMobject(r"Column Space of $A = x_{1}$",r"$\left( \begin{array}{c} 1 \\ 1 \\ 2 \\ 3 \end{array} \right)$",r"$+x_{2}$",r"$ \left( \begin{array}{c} 2 \\ 3 \\ 1 \\ 2 \end{array} \right)$",r"$ + x_{3}$",r"$\left( \begin{array}{c} 1 \\ 1 \\ 4 \\ 3 \end{array} \right)$") + CS_A.move_to(1.5*LEFT+1*DOWN) + CS_A[1].set_color(color = DARK_BLUE) + CS_A[3].set_color(color = DARK_BLUE) + CS_A[5].set_color(color = DARK_BLUE) + CS_A.scale(0.75) + + self.play(Write(CS_A),run_time = 2) + + arrow1 = Arrow(start = 1.25*UP,end = 0.25*DOWN+1.75*LEFT) + arrow2 = Arrow(start = 1.35*UP+0.5*RIGHT,end = 0.25*DOWN+0.5*RIGHT) + arrow3 = Arrow(start = 1.25*UP+0.75*RIGHT,end = 0.25*DOWN+2.9*RIGHT) + + Defn = TextMobject("Linear Combination of Columns of Matrix") + Defn.move_to(3*DOWN) + + self.play(Write(Defn), ShowCreation(arrow1), ShowCreation(arrow2), ShowCreation(arrow3),run_time = 1) + self.wait(1)
\ No newline at end of file diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file1_Axb.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file1_Axb.py new file mode 100755 index 0000000..95d1021 --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file1_Axb.py @@ -0,0 +1,77 @@ +from manimlib.imports import *
+
+class Axb(Scene):
+
+ def construct(self):
+
+ text0 = TextMobject("Linear System Of Equations")
+ text1 = TextMobject(r"$x_{1}+x_{2}+x_{3} =b_{1}$")
+ text2 = TextMobject(r"$x_{1}+2x_{2}+x_{3} =b_{2}$")
+ text3 = TextMobject(r"$x_{1}+x_{2}+3x_{3} =b_{3}$")
+ text0.move_to(UP*2+LEFT*2)
+ text0.set_color(DARK_BLUE)
+ text1.move_to(UP)
+ text2.move_to(ORIGIN)
+ text3.move_to(DOWN)
+
+ text0.scale(0.75)
+ text1.scale(0.75)
+ text2.scale(0.75)
+ text3.scale(0.75)
+ self.play(Write(text0))
+ self.play(Write(text1))
+ self.play(Write(text2))
+ self.play(Write(text3))
+ self.play(ApplyMethod(text0.move_to,3*UP+LEFT*2), ApplyMethod(text1.move_to,2.5*UP), ApplyMethod(text2.move_to,2*UP), ApplyMethod(text3.move_to,1.5*UP))
+
+ A = TextMobject(r"$\left( \begin{array}{c c c} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 3 \end{array}\right) \left[ \begin{array} {c} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] =$", r"$\left[ \begin{array}{c} x_{1}+x_{2}+x_{3} \\ x_{1}+2x_{2}+x_{3} \\ x_{1}+x_{2}+3x_{3} \end{array}\right]$")
+ A.scale(0.75)
+ self.play(FadeIn(A))
+
+ textA = TextMobject("A")
+ textx = TextMobject("x")
+ textb = TextMobject("Ax")
+
+ textA.move_to(DOWN+3*LEFT)
+ textx.move_to(1.1*DOWN+0.5*LEFT)
+ textb.move_to(DOWN-2*LEFT)
+
+ self.play(Write(textA), Write(textx), Write(textb))
+
+ circle1 = Circle(radius = 0.24)
+ circle2 = Circle(radius = 0.24)
+ square = Square(side_length = 0.6)
+
+ circle1.move_to(UP*0.5+LEFT*3.05)
+ circle2.move_to(UP*0.4+LEFT*0.5)
+ square.move_to(UP*0.4+RIGHT*1.3)
+
+ self.play(FadeIn(circle1), FadeIn(circle2),FadeIn(square))
+
+ self.play(ApplyMethod(circle1.move_to,UP*0.5+LEFT*2.45), ApplyMethod(circle2.move_to,LEFT*0.5), ApplyMethod(square.move_to,UP*0.4+RIGHT*2.2))
+ self.play(ApplyMethod(circle1.move_to,UP*0.5+LEFT*1.85), ApplyMethod(circle2.move_to,DOWN*0.5+LEFT*0.5), ApplyMethod(square.move_to,UP*0.4+RIGHT*3.1))
+
+ self.play(ApplyMethod(circle1.move_to,LEFT*3.05), ApplyMethod(circle2.move_to,UP*0.4+LEFT*0.5), ApplyMethod(square.move_to,RIGHT*1.3))
+ self.play(ApplyMethod(circle1.move_to,LEFT*2.45), ApplyMethod(circle2.move_to,LEFT*0.5), ApplyMethod(square.move_to,RIGHT*2.2))
+ self.play(ApplyMethod(circle1.move_to,LEFT*1.85), ApplyMethod(circle2.move_to,DOWN*0.5+LEFT*0.5), ApplyMethod(square.move_to,RIGHT*3.1))
+
+ self.play(ApplyMethod(circle1.move_to,0.4*DOWN+LEFT*3.05), ApplyMethod(circle2.move_to,UP*0.4+LEFT*0.5), ApplyMethod(square.move_to,0.4*DOWN+RIGHT*1.3))
+ self.play(ApplyMethod(circle1.move_to,0.4*DOWN+LEFT*2.45), ApplyMethod(circle2.move_to,LEFT*0.5), ApplyMethod(square.move_to,0.4*DOWN+RIGHT*2.2))
+ self.play(ApplyMethod(circle1.move_to,0.4*DOWN+LEFT*1.85), ApplyMethod(circle2.move_to,DOWN*0.5+LEFT*0.5), ApplyMethod(square.move_to,0.4*DOWN+RIGHT*3.1))
+
+ self.play(FadeOut(circle1), FadeOut(circle2), FadeOut(square))
+ self.play(FadeOut(A[0]), ApplyMethod(A[1].move_to,2*LEFT),ApplyMethod(textb.move_to,DOWN+1.7*LEFT), FadeOut(textx), FadeOut(textA))
+ b = TextMobject(r"$=\left[ \begin{array}{c} b_{1} \\ b_{2} \\ b_{3} \end{array}\right]$")
+ b.move_to(RIGHT)
+ textB = TextMobject("b")
+ textB.move_to(1.2*DOWN+1.1*RIGHT)
+ self.play(FadeIn(b),FadeIn(textB))
+
+ self.wait()
+
+ self.play(FadeOut(text0), FadeOut(text1), FadeOut(text2), FadeOut(text3))
+
+ axb = TextMobject("Ax = b")
+ self.play(FadeIn(axb), FadeOut(textb), FadeOut(textB), FadeOut(b), FadeOut(A[1]))
+
+ self.wait()
\ No newline at end of file diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file2_CSasImage.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file2_CSasImage.py new file mode 100644 index 0000000..70547cb --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file2_CSasImage.py @@ -0,0 +1,169 @@ +from manimlib.imports import * + +class Column_Space(Scene): + def construct(self): + + A = TextMobject(r"$A = $",r"$\left( \begin{array}{c c} 1 & 2 \\ 3 & 4 \end{array} \right)$") + A.move_to(2*UP) + A[1].set_color(color = DARK_BLUE) + A.scale(0.75) + + self.play(Write(A),run_time = 1) + + CS_A = TextMobject(r"Column Space of $A = x_{1}$",r"$\left( \begin{array}{c} 1 \\ 3 \end{array} \right)$",r"$+x_{2}$",r"$ \left( \begin{array}{c} 2 \\ 4\end{array} \right)$") + CS_A.move_to(1.5*LEFT+1*DOWN) + CS_A[1].set_color(color = DARK_BLUE) + CS_A[3].set_color(color = DARK_BLUE) + CS_A.scale(0.75) + + self.play(Write(CS_A),run_time = 2) + + arrow1 = Arrow(start = 1.25*UP,end = (0.25*DOWN+1.75*LEFT+0.25*DOWN+1.2*RIGHT)/2) + arrow3 = Arrow(start = 1.25*UP+0.75*RIGHT,end = (0.25*DOWN+2.9*RIGHT+0.25*DOWN)/2) + + arrow1.scale(1.5) + arrow3.scale(1.5) + + Defn = TextMobject("Linear Combination of Columns of Matrix") + Defn.move_to(3*DOWN) + + self.play(Write(Defn), ShowCreation(arrow1), ShowCreation(arrow3),run_time = 1) + self.wait(1) + +class solution(LinearTransformationScene): + def construct(self): + + self.setup() + self.wait() + + o = TextMobject(r"Consider the vector space $R^2$") + o.move_to(2*DOWN) + o.scale(0.75) + o.add_background_rectangle() + self.play(Write(o)) + self.wait() + self.play(FadeOut(o)) + + A = TextMobject(r"Let $A$ be ",r"$\left[\begin{array}{c c} 1 & -1 \\ 1 & -1 \end{array}\right]$",r". $A$ denotes the matrix the of this linear transformation.") + A.move_to(2*DOWN) + A.scale(0.75) + A.add_background_rectangle() + self.play(Write(A)) + matrix = [[1,-1],[1,-1]] + self.apply_matrix(matrix) + self.wait() + self.play(FadeOut(A)) + + o = TextMobject(r"This is the transformed vector space") + o.move_to(2*DOWN) + o.scale(0.75) + o.add_background_rectangle() + self.play(Write(o)) + self.wait() + self.play(FadeOut(o)) + + texti = TextMobject(r"$\left[\begin{array}{c}1\\1\end{array}\right]$") + textj = TextMobject(r"$\left[\begin{array}{c}-1\\-1\end{array}\right]$") + texti.set_color(GREEN) + textj.set_color(RED) + texti.scale(0.7) + textj.scale(0.7) + texti.move_to(1.35*RIGHT+0.5*UP) + textj.move_to(-(1.5*RIGHT+0.5*UP)) + + text1 = TextMobject("[") + text2 = TextMobject(r"$\begin{array}{c} 1 \\ 1 \end{array}$") + text3 = TextMobject(r"$\begin{array}{c} -1 \\ -1 \end{array}$") + text4 = TextMobject("]") + + text2.set_color(GREEN) + text3.set_color(RED) + + text1.scale(2) + text4.scale(2) + text2.scale(0.7) + text3.scale(0.7) + + text1.move_to(2.5*UP+6*LEFT) + text2.move_to(2.5*UP+5.75*LEFT) + text3.move_to(2.5*UP+5.25*LEFT) + text4.move_to(2.5*UP+5*LEFT) + + self.play(Write(texti), Write(textj)) + self.wait() + self.play(FadeIn(text1), Transform(texti,text2), Transform(textj,text3), FadeIn(text4)) + self.wait() + + o = TextMobject(r"Now, you can observe the Image of Linear Transformation") + o1 = TextMobject(r"and Column Space(i.e. span of columns of matrix $A$) are same") + o.move_to(2.5*DOWN) + o1.move_to(3*DOWN) + o.scale(0.75) + o1.scale(0.75) + o.add_background_rectangle() + o1.add_background_rectangle() + self.play(Write(o)) + self.play(Write(o1)) + self.wait() + self.play(FadeOut(o),FadeOut(o1)) + +class solution2nd(LinearTransformationScene): + def construct(self): + + self.setup() + self.wait() + + arrow1 = Arrow(start = ORIGIN,end = 2*DOWN+RIGHT) + arrow2 = Arrow(start = ORIGIN,end = UP+LEFT) + arrow3 = Arrow(start = ORIGIN,end = 3*UP+4*RIGHT) + arrow1.set_color(YELLOW) + arrow2.set_color(ORANGE) + arrow3.set_color(PURPLE) + arrow1.scale(1.3) + arrow2.scale(1.5) + arrow3.scale(1.1) + + self.play(ShowCreation(arrow1), ShowCreation(arrow2), ShowCreation(arrow3)) + + self.add_transformable_mobject(arrow1) + self.add_transformable_mobject(arrow2) + self.add_transformable_mobject(arrow3) + o = TextMobject(r"Consider any vector in the original vector space $R^2$") + o.move_to(2.5*DOWN) + o.scale(0.75) + o.add_background_rectangle() + self.play(Write(o)) + self.wait() + self.play(FadeOut(o)) + + A = TextMobject(r"Let the matrix the of this linear transformation be $A$ =",r"$\left[\begin{array}{c c} 1 & -1 \\ 1 & -1 \end{array}\right]$",r" again.") + A.move_to(2*DOWN) + A.scale(0.75) + A.add_background_rectangle() + self.play(Write(A)) + matrix = [[1,-1],[1,-1]] + self.apply_matrix(matrix) + self.wait() + self.play(FadeOut(A)) + + o = TextMobject(r"This is the transformed vector space") + o.move_to(2*DOWN) + o.scale(0.75) + o.add_background_rectangle() + self.play(Write(o)) + self.wait() + self.play(FadeOut(o)) + + o = TextMobject(r"Each and every vector of original vector space $R^2$ will transform") + o1 = TextMobject(r"to this new vector space which is spanned by $\mathbf{CS}(A)$") + o.move_to(2.5*DOWN) + o1.move_to(3*DOWN) + o.scale(0.75) + o1.scale(0.75) + o.add_background_rectangle() + o1.add_background_rectangle() + self.play(Write(o)) + self.play(Write(o1)) + self.wait() + self.play(FadeOut(o)) + self.play(FadeOut(o1))
\ No newline at end of file diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file3_solution.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file3_solution.py new file mode 100644 index 0000000..eb310f3 --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file3_solution.py @@ -0,0 +1,77 @@ +from manimlib.imports import * +class solution(LinearTransformationScene): + def construct(self): + + self.setup() + self.wait() + + o = TextMobject(r"This is the original vector space $R^2$ (before Linear Transformation)") + o.move_to(3*DOWN) + o.scale(0.75) + o.add_background_rectangle() + self.play(Write(o)) + self.wait() + self.play(FadeOut(o)) + + A = TextMobject(r"Consider the matrix the of this linear transformation $A$ = $\left[\begin{array}{c c} 1 & -1 \\ 1 & -1 \end{array}\right]$") + A.move_to(3*DOWN) + A.scale(0.75) + A.add_background_rectangle() + self.play(Write(A)) + matrix = [[1,-1],[1,-1]] + self.apply_matrix(matrix) + self.wait() + self.play(FadeOut(A)) + + o = TextMobject(r"This is the transformed vector space") + o.move_to(3*DOWN) + o.scale(0.75) + o.add_background_rectangle() + self.play(Write(o)) + self.wait() + self.play(FadeOut(o)) + + arrow2 = Arrow(start = ORIGIN, end = 2*DOWN+2*LEFT) + arrow2.set_color(PURPLE) + arrow2.scale(1.2) + self.play(ShowCreation(arrow2)) + self.wait() + + o1 = TextMobject("If the ","vector b"," lies in the transformed vector space") + o2 = TextMobject("(the line) then the solution exist") + o1.move_to(2.5*DOWN+2*RIGHT) + o1[1].set_color(PURPLE) + o2.move_to(3*DOWN+2.5*RIGHT) + o1.scale(0.75) + o2.scale(0.75) + o1.add_background_rectangle() + o2.add_background_rectangle() + self.play(Write(o1)) + self.play(Write(o2)) + self.wait() + self.play(FadeOut(o1), FadeOut(o2)) + + self.play(FadeOut(arrow2)) + + arrow1 = Arrow(start = ORIGIN, end = 2*UP+RIGHT) + arrow1.set_color(ORANGE) + arrow1.scale(1.3) + self.play(ShowCreation(arrow1)) + self.wait() + + o1 = TextMobject("If the ","vector b"," does lies in the transformed") + o2 = TextMobject("vector space then the does not solution exist") + o1.move_to(2.5*DOWN+2*RIGHT) + o1[1].set_color(ORANGE) + o2.move_to(3*DOWN+2.5*RIGHT) + o1.scale(0.75) + o2.scale(0.75) + o1.add_background_rectangle() + o2.add_background_rectangle() + self.play(Write(o1)) + self.play(Write(o2)) + self.wait() + self.play(FadeOut(o1), FadeOut(o2)) + + self.play(FadeOut(arrow1)) +
\ No newline at end of file diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file4_null_space.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file4_null_space.py new file mode 100644 index 0000000..3c52677 --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file4_null_space.py @@ -0,0 +1,91 @@ +from manimlib.imports import * +class null_space(LinearTransformationScene): + def construct(self): + + self.setup() + self.wait() + + o = TextMobject(r"This is the original vector space $R^2$(before Linear Transformation)") + o.move_to(3*DOWN) + o.scale(0.75) + o.add_background_rectangle() + self.play(Write(o)) + self.wait() + self.play(FadeOut(o)) + + o1 = TextMobject("Consider a set of vectors which are linear") + o2 = TextMobject(r"span of a particular vector $\left(\begin{array}{c} 1 \\ 1 \end{array}\right)$") + o1.move_to(2*DOWN+3*RIGHT) + o2.move_to(2.75*DOWN+3*RIGHT) + o1.scale(0.7) + o2.scale(0.7) + o1.add_background_rectangle() + o2.add_background_rectangle() + self.play(Write(o1)) + self.play(Write(o2)) + + arrow = Arrow(start = ORIGIN, end = UP+RIGHT) + arrow1 = Arrow(start = ORIGIN, end = 2*(UP+RIGHT)) + arrow2 = Arrow(start = ORIGIN, end = 3*(UP+RIGHT)) + arrow3 = Arrow(start = ORIGIN, end = 4*(UP+RIGHT)) + arrow4 = Arrow(start = ORIGIN, end = DOWN+LEFT) + arrow5 = Arrow(start = ORIGIN, end = 2*(DOWN+LEFT)) + arrow6 = Arrow(start = ORIGIN, end = 3*(DOWN+LEFT)) + arrow7 = Arrow(start = ORIGIN, end = 4*(DOWN+LEFT)) + + arrow.scale(1.5) + arrow1.scale(1.2) + arrow2.scale(1.15) + arrow3.scale(1.1) + arrow4.scale(1.5) + arrow5.scale(1.2) + arrow6.scale(1.15) + arrow7.scale(1.1) + + self.play(ShowCreation(arrow), + ShowCreation(arrow1), + ShowCreation(arrow2), + ShowCreation(arrow3), + ShowCreation(arrow4), + ShowCreation(arrow5), + ShowCreation(arrow6), + ShowCreation(arrow7), + ) + + self.wait(2) + self.play(FadeOut(o1), FadeOut(o2)) + + self.add_transformable_mobject(arrow) + self.add_transformable_mobject(arrow1) + self.add_transformable_mobject(arrow2) + self.add_transformable_mobject(arrow3) + self.add_transformable_mobject(arrow4) + self.add_transformable_mobject(arrow5) + self.add_transformable_mobject(arrow6) + self.add_transformable_mobject(arrow7) + + o1 = TextMobject("Notice, entire set of vectors which belongs to the vector") + o2 = TextMobject(r"subspace(Linear Span of $\left(\begin{array}{c} 1 \\ 1 \end{array}\right)$) transforms to zero") + o1.move_to(2*DOWN+2.5*RIGHT) + o2.move_to(2.75*DOWN+2.5*RIGHT) + o1.scale(0.7) + o2.scale(0.7) + o1.add_background_rectangle() + o2.add_background_rectangle() + self.play(Write(o1)) + self.play(Write(o2)) + self.wait() + + matrix = [[1,-1],[1,-1]] + self.apply_matrix(matrix) + self.wait() + + self.play(FadeOut(o1), FadeOut(o2)) + + o = TextMobject(r"Hence, the vector space formed by linear span of $\left(\begin{array}{c} 1 \\ 1 \end{array}\right)$ is the null space of $A$") + o.move_to(3*DOWN) + o.scale(0.75) + o.add_background_rectangle() + self.play(Write(o)) + self.wait(2) + self.play(FadeOut(o), FadeOut(arrow), FadeOut(arrow1), FadeOut(arrow2), FadeOut(arrow3), FadeOut(arrow4), FadeOut(arrow5), FadeOut(arrow6), FadeOut(arrow7)) diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file5_Row_Space_part_1.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file5_Row_Space_part_1.py new file mode 100644 index 0000000..5259eb4 --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file5_Row_Space_part_1.py @@ -0,0 +1,68 @@ +from manimlib.imports import * +class LS(Scene): + def construct(self): + text1 = TextMobject(r"Consider a matrix $A =$") + text2 = TextMobject(r"[") + text3 = TextMobject(r"$\begin{array}{c c} 1 & -2\end{array}$") + text4 = TextMobject(r"$\begin{array}{c c} 1 & -1\end{array}$") + text5 = TextMobject(r"]") + + text2.scale(2) + text5.scale(2) + + text1.set_color(DARK_BLUE) + text2.set_color(DARK_BLUE) + text3.set_color(PURPLE) + text4.set_color(YELLOW) + text5.set_color(DARK_BLUE) + + text1.move_to(3.5*LEFT+3*UP+2*RIGHT) + text2.move_to(0.75*LEFT+3*UP+2*RIGHT) + text3.move_to(3.25*UP+2*RIGHT) + text4.move_to(2.75*UP+2*RIGHT) + text5.move_to(0.75*RIGHT+3*UP+2*RIGHT) + + self.play(FadeIn(text1), FadeIn(text2), FadeIn(text3), FadeIn(text4), FadeIn(text5)) + self.wait() + + ttext1 = TextMobject(r"$A^T =$") + ttext2 = TextMobject(r"[") + ttext3 = TextMobject(r"$\begin{array}{c} 1 \\ -2\end{array}$") + ttext4 = TextMobject(r"$\begin{array}{c} 1 \\ -1\end{array}$") + ttext5 = TextMobject(r"]") + + ttext2.scale(2) + ttext5.scale(2) + + ttext1.set_color(DARK_BLUE) + ttext2.set_color(DARK_BLUE) + ttext3.set_color(PURPLE) + ttext4.set_color(YELLOW) + ttext5.set_color(DARK_BLUE) + + ttext1.move_to(2*LEFT+1.5*UP+2*RIGHT) + ttext2.move_to(1*LEFT+1.5*UP+2*RIGHT) + ttext3.move_to(0.5*LEFT+1.5*UP+2*RIGHT) + ttext4.move_to(0.5*RIGHT+1.5*UP+2*RIGHT) + ttext5.move_to(1*RIGHT+1.5*UP+2*RIGHT) + + self.play(FadeIn(ttext1), FadeIn(ttext2), FadeIn(ttext3), FadeIn(ttext4), FadeIn(ttext5)) + + rtext = TextMobject(r"Row Space of $A$ = Column Space of $A^T = a_1$",r"$\left[\begin{array}{c} 1 \\ -2\end{array}\right]$",r"$+a_2$",r"$\left[\begin{array}{c} 1 \\ -1\end{array}\right]$") + rtext[1].set_color(PURPLE) + rtext[3].set_color(YELLOW) + rtext.move_to(2*DOWN+1.5*LEFT) + rtext.scale(0.75) + + self.play(Write(rtext)) + self.wait() + + arrow1 = Arrow(start = 1.5*RIGHT+UP, end = 1.25*(DOWN+RIGHT)) + arrow2 = Arrow(start = 2.5*RIGHT+UP, end = 1.25*DOWN+3.25*RIGHT) + arrow1.scale(1.25) + arrow2.scale(1.25) + arrow1.set_color(PURPLE) + arrow2.set_color(YELLOW) + + self.play(ShowCreation(arrow1), ShowCreation(arrow2)) + self.wait(2) diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file6_Row_Space_part_2.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file6_Row_Space_part_2.py new file mode 100644 index 0000000..b16a32a --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file6_Row_Space_part_2.py @@ -0,0 +1,145 @@ +from manimlib.imports import * + +class Row_Space(Scene): + def construct(self): + + Heading = TextMobject("Row Space") + defn1 = TextMobject("Definition 1: Row Space of a matrix is the linear combination of the rows of that matrix.") + defn2 = TextMobject("Definition 2: It is a vector space generated by a linear combination of the columns of $A^{T}$.") + equivalent = TextMobject(r"Definition 1 $\equiv$ Definition 2") + + Heading.move_to(2*UP) + Heading.set_color(color = DARK_BLUE) + + defn1.move_to(UP) + defn1.scale(0.75) + + defn2.scale(0.75) + + equivalent.move_to(DOWN) + + self.play(Write(Heading)) + self.play(Write(defn1)) + self.play(Write(defn2)) + self.play(Write(equivalent)) + + self.wait(2) + self.play(FadeOut(Heading),FadeOut(defn1),FadeOut(defn2),ApplyMethod(equivalent.move_to,2*UP)) + + how = TextMobject("Let us see, How?") + how.move_to(UP) + self.play(Write(how)) + self.play(FadeOut(equivalent),FadeOut(how)) + + A = TextMobject(r"$A = $",r"$\left( \begin{array}{c c c} 1 & 2 & 1 \\ 1 & 3 & 1 \\ 2 & 1 & 4 \\ 3 & 2 & 3 \end{array} \right)$") + A.move_to(2*UP+3*LEFT) + A[1].set_color(color = DARK_BLUE) + A.scale(0.80) + + self.play(Write(A)) + + rows = TextMobject(r"Rows of A $\rightarrow$", + r"$\left( \begin{array}{c c c} 1 & 2 & 1 \end{array} \right)$,", + r"$ \left( \begin{array}{c c c} 1 & 3 & 1 \end{array} \right)$,", + r"$\left( \begin{array}{c c c} 2 & 1 & 4 \end{array} \right)$,", + r"$ \left( \begin{array}{c c c} 3 & 2 & 3 \end{array} \right)$") + rows.scale(0.75) + rows[1:5].set_color(DARK_BLUE) + self.play(Write(rows)) + + ac_defn1 = TextMobject("According to Definition 1 : ") + ac_defn1.move_to(DOWN) + + RS_A = TextMobject(r"Row Space of $A = x_{1}$", + r"$\left( \begin{array}{c c c} 1 & 2 & 1 \end{array} \right)$", + r"$+x_{2}$", + r"$ \left( \begin{array}{c c c} 1 & 3 & 1 \end{array} \right)$", + r"$ + x_{3}$", + r"$\left( \begin{array}{c c c} 2 & 1 & 4 \end{array} \right)$", + r"$+x_{4}$", + r"$ \left( \begin{array}{c c c} 3 & 2 & 3 \end{array} \right)$") + RS_A.move_to(DOWN+DOWN) + RS_A[6].move_to(2*DOWN+DOWN) + RS_A[7].move_to(2*DOWN+2*RIGHT+DOWN) + RS_A[1].set_color(color = DARK_BLUE) + RS_A[3].set_color(color = DARK_BLUE) + RS_A[5].set_color(color = DARK_BLUE) + RS_A[7].set_color(color = DARK_BLUE) + RS_A.scale(0.75) + + self.play(FadeOut(rows[0]),Transform(rows[1],RS_A[1]),Transform(rows[2],RS_A[3]),Transform(rows[3],RS_A[5]),Transform(rows[4],RS_A[7])) + self.play(FadeIn(ac_defn1), Write(RS_A)) + self.wait(1) + + self.play(FadeOut(rows[1]), FadeOut(rows[2]), FadeOut(rows[3]), FadeOut(rows[4]), FadeOut(RS_A), FadeOut(ac_defn1)) + + A_T = TextMobject(r"$A^{T} = $",r"$\left( \begin{array}{c c c c} 1 & 1 & 2 & 3 \\ 2 & 3 & 1 & 2 \\ 1 & 1 & 4 & 3 \end{array} \right)$") + A_T.move_to(2*UP+3*RIGHT) + A_T[1].set_color(color = DARK_BLUE) + A_T.scale(0.80) + + self.play(Write(A_T)) + + change1 = TextMobject(r"Rows of $A\equiv$ Columns of $A^{T}$") + change2 = TextMobject(r"Columns of $A\equiv$ Rows of $A^{T}$") + change2.move_to(DOWN) + + change3 = TextMobject(r"Row Space of $A$ = Linear Combination of",r"Rows","of",r"A") + change3.move_to(2*DOWN) + change3[1].set_color(DARK_BLUE) + change3[3].set_color(DARK_BLUE) + + self.play(Write(change1)) + self.play(Write(change2)) + self.play(Write(change3)) + + columns = TextMobject("Columns") + columns.scale(0.6) + columns.set_color(DARK_BLUE) + columns.move_to(2*DOWN+4.1*RIGHT) + + a = TextMobject(r"$A^{T}$") + a.set_color(DARK_BLUE) + a.move_to(1.95*DOWN+5.6*RIGHT) + + self.wait(0.5) + + self.play(Transform(change3[1],columns), Transform(change3[3],a)) + + equal = TextMobject(r"= Column Space($A^{T}$)") + equal.move_to(3*DOWN+0.5*RIGHT) + + self.play(Write(equal)) + + self.play(FadeOut(A_T), FadeOut(change1), FadeOut(change2), FadeOut(change3), FadeOut(A), FadeOut(equal)) + + ac_defn1.move_to(3*UP) + RS_A.move_to(1.5*UP) + RS_A[6].move_to(UP) + RS_A[7].move_to(UP+1.5*RIGHT) + + self.play(Write(RS_A),FadeIn(ac_defn1)) + + CS_AT = TextMobject(r"Row Space of $A = x_{1}$", + r"$\left( \begin{array}{c} 1 \\ 2 \\ 1 \end{array} \right)$", + r"$+x_{2}$", + r"$ \left( \begin{array}{c} 1 \\ 3 \\ 1 \end{array} \right)$", + r"$ + x_{3}$", + r"$\left( \begin{array}{c} 2 \\ 1 \\ 4 \end{array} \right)$", + r"$+x_{4}$", + r"$ \left( \begin{array}{c} 3 \\ 2 \\ 3 \end{array} \right)$") + CS_AT.move_to(1.5*DOWN) + CS_AT[1].set_color(color = DARK_BLUE) + CS_AT[3].set_color(color = DARK_BLUE) + CS_AT[5].set_color(color = DARK_BLUE) + CS_AT[7].set_color(color = DARK_BLUE) + CS_AT.scale(0.75) + + ac_defn2 = TextMobject("According to Definition 2 : ") + equivalent = TextMobject(r"Hence, Definition 1 $\equiv$ Definition 2") + equivalent.move_to(3*DOWN) + + self.play(Write(CS_AT),FadeIn(ac_defn2)) + self.play(Write(equivalent)) + + self.wait() diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file7_Row_space_Orthogonal_Complements.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file7_Row_space_Orthogonal_Complements.py new file mode 100644 index 0000000..c81d370 --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file7_Row_space_Orthogonal_Complements.py @@ -0,0 +1,150 @@ +from manimlib.imports import * +class row_space(LinearTransformationScene): + def construct(self): + + self.setup() + self.wait() + + o = TextMobject(r"This is the original vector space $R^2$(before Linear Transformation)") + o.move_to(DOWN) + o.scale(0.75) + o.add_background_rectangle() + self.play(Write(o)) + self.wait() + self.play(FadeOut(o)) + + o1 = TextMobject("Consider a set of vectors which are linear") + o2 = TextMobject(r"span of a $\left(\begin{array}{c} 1 \\ 1 \end{array}\right)$i.e. the null space.") + o1.move_to(2*DOWN+3*RIGHT) + o2.move_to(2.75*DOWN+3*RIGHT) + o1.scale(0.7) + o2.scale(0.7) + o1.add_background_rectangle() + o2.add_background_rectangle() + self.play(Write(o1)) + self.play(Write(o2)) + + arrow = Arrow(start = ORIGIN, end = UP+RIGHT) + arrow.set_color(YELLOW) + arrow1 = Arrow(start = ORIGIN, end = 2*(UP+RIGHT)) + arrow1.set_color(YELLOW) + arrow2 = Arrow(start = ORIGIN, end = 3*(UP+RIGHT)) + arrow2.set_color(YELLOW) + arrow3 = Arrow(start = ORIGIN, end = 4*(UP+RIGHT)) + arrow3.set_color(YELLOW) + arrow4 = Arrow(start = ORIGIN, end = DOWN+LEFT) + arrow4.set_color(YELLOW) + arrow5 = Arrow(start = ORIGIN, end = 2*(DOWN+LEFT)) + arrow5.set_color(YELLOW) + arrow6 = Arrow(start = ORIGIN, end = 3*(DOWN+LEFT)) + arrow6.set_color(YELLOW) + arrow7 = Arrow(start = ORIGIN, end = 4*(DOWN+LEFT)) + arrow7.set_color(YELLOW) + + arrow.scale(1.5) + arrow1.scale(1.2) + arrow2.scale(1.15) + arrow3.scale(1.1) + arrow4.scale(1.5) + arrow5.scale(1.2) + arrow6.scale(1.15) + arrow7.scale(1.1) + + self.play(ShowCreation(arrow), + ShowCreation(arrow1), + ShowCreation(arrow2), + ShowCreation(arrow3), + ShowCreation(arrow4), + ShowCreation(arrow5), + ShowCreation(arrow6), + ShowCreation(arrow7), + ) + + self.wait(2) + self.play(FadeOut(o1), FadeOut(o2)) + + o1 = TextMobject("Consider a set of vectors which are linear") + o2 = TextMobject(r"span of a $\left(\begin{array}{c} 1 \\ -1 \end{array}\right)$i.e. the row space.") + o1.move_to(2*DOWN+3*RIGHT) + o2.move_to(2.75*DOWN+3*RIGHT) + o1.scale(0.7) + o2.scale(0.7) + o1.add_background_rectangle() + o2.add_background_rectangle() + self.play(Write(o1)) + self.play(Write(o2)) + + rarrow = Arrow(start = ORIGIN, end = -UP+RIGHT) + rarrow.set_color(PURPLE) + rarrow1 = Arrow(start = ORIGIN, end = 2*(-UP+RIGHT)) + rarrow1.set_color(PURPLE) + rarrow2 = Arrow(start = ORIGIN, end = 3*(-UP+RIGHT)) + rarrow2.set_color(PURPLE) + rarrow3 = Arrow(start = ORIGIN, end = 4*(-UP+RIGHT)) + rarrow3.set_color(PURPLE) + rarrow4 = Arrow(start = ORIGIN, end = -DOWN+LEFT) + rarrow4.set_color(PURPLE) + rarrow5 = Arrow(start = ORIGIN, end = 2*(-DOWN+LEFT)) + rarrow5.set_color(PURPLE) + rarrow6 = Arrow(start = ORIGIN, end = 3*(-DOWN+LEFT)) + rarrow6.set_color(PURPLE) + rarrow7 = Arrow(start = ORIGIN, end = 4*(-DOWN+LEFT)) + rarrow7.set_color(PURPLE) + + rarrow.scale(1.5) + rarrow1.scale(1.2) + rarrow2.scale(1.15) + rarrow3.scale(1.1) + rarrow4.scale(1.5) + rarrow5.scale(1.2) + rarrow6.scale(1.15) + rarrow7.scale(1.1) + + self.play(ShowCreation(rarrow), + ShowCreation(rarrow1), + ShowCreation(rarrow2), + ShowCreation(rarrow3), + ShowCreation(rarrow4), + ShowCreation(rarrow5), + ShowCreation(rarrow6), + ShowCreation(rarrow7), + ) + + self.wait(2) + self.play(FadeOut(o1), FadeOut(o2)) + + self.add_transformable_mobject(arrow) + self.add_transformable_mobject(arrow1) + self.add_transformable_mobject(arrow2) + self.add_transformable_mobject(arrow3) + self.add_transformable_mobject(arrow4) + self.add_transformable_mobject(arrow5) + self.add_transformable_mobject(arrow6) + self.add_transformable_mobject(arrow7) + + self.add_transformable_mobject(rarrow) + self.add_transformable_mobject(rarrow1) + self.add_transformable_mobject(rarrow2) + self.add_transformable_mobject(rarrow3) + self.add_transformable_mobject(rarrow4) + self.add_transformable_mobject(rarrow5) + self.add_transformable_mobject(rarrow6) + self.add_transformable_mobject(rarrow7) + + o1 = TextMobject("Notice, entire set of vectors which belong to the null space of $A$ transforms to zero") + o2 = TextMobject(r"and entire set of vectors which belong to the row space of $A$ transforms to column space of $A$.") + o1.move_to(2.5*DOWN) + o2.move_to(3.5*DOWN) + o1.scale(0.7) + o2.scale(0.7) + o1.add_background_rectangle() + o2.add_background_rectangle() + self.play(Write(o1)) + self.play(Write(o2)) + self.wait() + + matrix = [[1,-1],[1,-1]] + self.apply_matrix(matrix) + self.wait(3) + + self.play(FadeOut(o1), FadeOut(o2))
\ No newline at end of file diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file8_Left_Null_Space.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file8_Left_Null_Space.py new file mode 100755 index 0000000..fd05e75 --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file8_Left_Null_Space.py @@ -0,0 +1,26 @@ +from manimlib.imports import * + +class Left_Null_Space(Scene): + def construct(self): + + A = TextMobject(r"Left Null Space of A") + A.move_to(3*UP) + defn = TextMobject(r"It is a vector space that consists of all the solution $x$ to the equation $A^{T}x=0$") + defn.move_to(2*UP) + defn.scale(0.75) + eqn1 = TextMobject(r"$A^{T}x=0 \cdots (i)$") + eqn1.move_to(UP) + self.play(Write(A), Write(defn), Write(eqn1),run_time=1) + statement = TextMobject(r"Taking transpose of eqn $(i)$") + eqn = TextMobject(r"$(A^{T}x)^{T}=0$") + eqn.move_to(DOWN) + eqn2 = TextMobject(r"$x^{T}(A^{T})^{T}=0$") + eqn2.move_to(DOWN) + eqn3 = TextMobject(r"$x^{T}A=0$") + eqn3.move_to(DOWN) + self.play(Write(statement),Write(eqn),run_time=1) + self.wait(0.5) + self.play(Transform(eqn,eqn2),run_time=1) + self.wait(0.5) + self.play(Transform(eqn,eqn3),run_time=1) + self.wait(0.5)
\ No newline at end of file diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file9_left_null_space.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file9_left_null_space.py new file mode 100644 index 0000000..61285be --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file9_left_null_space.py @@ -0,0 +1,186 @@ +from manimlib.imports import * +class row_space(LinearTransformationScene): + def construct(self): + + self.setup() + self.wait() + + o = TextMobject(r"This is the original vector space $R^2$(before Linear Transformation)") + o.move_to(DOWN) + o.scale(0.75) + o.add_background_rectangle() + self.play(Write(o)) + self.wait() + self.play(FadeOut(o)) + + o1 = TextMobject("Consider a set of vectors which are linear") + o2 = TextMobject(r"span of a $\left(\begin{array}{c} 1 \\ 1 \end{array}\right)$i.e. the null space.") + o1.move_to(2*DOWN+3*RIGHT) + o2.move_to(2.75*DOWN+3*RIGHT) + o1.scale(0.7) + o2.scale(0.7) + o1.add_background_rectangle() + o2.add_background_rectangle() + self.play(Write(o1)) + self.play(Write(o2)) + + arrow = Arrow(start = ORIGIN, end = UP+RIGHT) + arrow.set_color(YELLOW) + arrow1 = Arrow(start = ORIGIN, end = 2*(UP+RIGHT)) + arrow1.set_color(YELLOW) + arrow2 = Arrow(start = ORIGIN, end = 3*(UP+RIGHT)) + arrow2.set_color(YELLOW) + arrow3 = Arrow(start = ORIGIN, end = 4*(UP+RIGHT)) + arrow3.set_color(YELLOW) + arrow4 = Arrow(start = ORIGIN, end = DOWN+LEFT) + arrow4.set_color(YELLOW) + arrow5 = Arrow(start = ORIGIN, end = 2*(DOWN+LEFT)) + arrow5.set_color(YELLOW) + arrow6 = Arrow(start = ORIGIN, end = 3*(DOWN+LEFT)) + arrow6.set_color(YELLOW) + arrow7 = Arrow(start = ORIGIN, end = 4*(DOWN+LEFT)) + arrow7.set_color(YELLOW) + + arrow.scale(1.5) + arrow1.scale(1.2) + arrow2.scale(1.15) + arrow3.scale(1.1) + arrow4.scale(1.5) + arrow5.scale(1.2) + arrow6.scale(1.15) + arrow7.scale(1.1) + + self.play(ShowCreation(arrow), + ShowCreation(arrow1), + ShowCreation(arrow2), + ShowCreation(arrow3), + ShowCreation(arrow4), + ShowCreation(arrow5), + ShowCreation(arrow6), + ShowCreation(arrow7), + ) + + self.wait(2) + self.play(FadeOut(o1), FadeOut(o2)) + + o1 = TextMobject("Consider a set of vectors which are linear") + o2 = TextMobject(r"span of a vector $\left(\begin{array}{c} 1 \\ -1 \end{array}\right)$i.e. the row space.") + o1.move_to(2*DOWN+3*RIGHT) + o2.move_to(2.75*DOWN+3*RIGHT) + o1.scale(0.7) + o2.scale(0.7) + o1.add_background_rectangle() + o2.add_background_rectangle() + self.play(Write(o1)) + self.play(Write(o2)) + + rarrow = Arrow(start = ORIGIN, end = -UP+RIGHT) + rarrow.set_color(PURPLE) + rarrow1 = Arrow(start = ORIGIN, end = 2*(-UP+RIGHT)) + rarrow1.set_color(PURPLE) + rarrow2 = Arrow(start = ORIGIN, end = 3*(-UP+RIGHT)) + rarrow2.set_color(PURPLE) + rarrow3 = Arrow(start = ORIGIN, end = 4*(-UP+RIGHT)) + rarrow3.set_color(PURPLE) + rarrow4 = Arrow(start = ORIGIN, end = -DOWN+LEFT) + rarrow4.set_color(PURPLE) + rarrow5 = Arrow(start = ORIGIN, end = 2*(-DOWN+LEFT)) + rarrow5.set_color(PURPLE) + rarrow6 = Arrow(start = ORIGIN, end = 3*(-DOWN+LEFT)) + rarrow6.set_color(PURPLE) + rarrow7 = Arrow(start = ORIGIN, end = 4*(-DOWN+LEFT)) + rarrow7.set_color(PURPLE) + + rarrow.scale(1.5) + rarrow1.scale(1.2) + rarrow2.scale(1.15) + rarrow3.scale(1.1) + rarrow4.scale(1.5) + rarrow5.scale(1.2) + rarrow6.scale(1.15) + rarrow7.scale(1.1) + + self.play(ShowCreation(rarrow), + ShowCreation(rarrow1), + ShowCreation(rarrow2), + ShowCreation(rarrow3), + ShowCreation(rarrow4), + ShowCreation(rarrow5), + ShowCreation(rarrow6), + ShowCreation(rarrow7), + ) + + self.wait(2) + self.play(FadeOut(o1), FadeOut(o2)) + + self.add_transformable_mobject(arrow) + self.add_transformable_mobject(arrow1) + self.add_transformable_mobject(arrow2) + self.add_transformable_mobject(arrow3) + self.add_transformable_mobject(arrow4) + self.add_transformable_mobject(arrow5) + self.add_transformable_mobject(arrow6) + self.add_transformable_mobject(arrow7) + + self.add_transformable_mobject(rarrow) + self.add_transformable_mobject(rarrow1) + self.add_transformable_mobject(rarrow2) + self.add_transformable_mobject(rarrow3) + self.add_transformable_mobject(rarrow4) + self.add_transformable_mobject(rarrow5) + self.add_transformable_mobject(rarrow6) + self.add_transformable_mobject(rarrow7) + + matrix = [[1,-1],[1,-1]] + self.apply_matrix(matrix) + self.wait(3) + + o1 = TextMobject("Consider a set of vectors which are linear span of a vector") + o2 = TextMobject(r"$\left(\begin{array}{c} 1 \\ -1 \end{array}\right)$ which is orthogonal to column space i.e. Left Null Space") + o1.move_to(2*DOWN) + o2.move_to(2.75*DOWN) + o1.scale(0.7) + o2.scale(0.7) + o1.add_background_rectangle() + o2.add_background_rectangle() + self.play(Write(o1)) + self.play(Write(o2)) + + rarrow = Arrow(start = ORIGIN, end = -UP+RIGHT) + rarrow.set_color(YELLOW) + rarrow1 = Arrow(start = ORIGIN, end = 2*(-UP+RIGHT)) + rarrow1.set_color(YELLOW) + rarrow2 = Arrow(start = ORIGIN, end = 3*(-UP+RIGHT)) + rarrow2.set_color(YELLOW) + rarrow3 = Arrow(start = ORIGIN, end = 4*(-UP+RIGHT)) + rarrow3.set_color(YELLOW) + rarrow4 = Arrow(start = ORIGIN, end = -DOWN+LEFT) + rarrow4.set_color(YELLOW) + rarrow5 = Arrow(start = ORIGIN, end = 2*(-DOWN+LEFT)) + rarrow5.set_color(YELLOW) + rarrow6 = Arrow(start = ORIGIN, end = 3*(-DOWN+LEFT)) + rarrow6.set_color(YELLOW) + rarrow7 = Arrow(start = ORIGIN, end = 4*(-DOWN+LEFT)) + rarrow7.set_color(YELLOW) + + rarrow.scale(1.5) + rarrow1.scale(1.2) + rarrow2.scale(1.15) + rarrow3.scale(1.1) + rarrow4.scale(1.5) + rarrow5.scale(1.2) + rarrow6.scale(1.15) + rarrow7.scale(1.1) + + self.play(ShowCreation(rarrow), + ShowCreation(rarrow1), + ShowCreation(rarrow2), + ShowCreation(rarrow3), + ShowCreation(rarrow4), + ShowCreation(rarrow5), + ShowCreation(rarrow6), + ShowCreation(rarrow7), + ) + + self.wait(2) + self.play(FadeOut(o1), FadeOut(o2)) |
