diff options
Diffstat (limited to 'FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces')
4 files changed, 411 insertions, 0 deletions
diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/Axb.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/Axb.py new file mode 100755 index 0000000..95d1021 --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/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/CSasImage.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/CSasImage.py new file mode 100644 index 0000000..fbb3291 --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/CSasImage.py @@ -0,0 +1,168 @@ +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) + +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 $A$(= ",r"$\left[\begin{array}{c c} 1 & -1 \\ 1 & -1 \end{array}\right]$",r")denote 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(YELLOW) + arrow3.set_color(YELLOW) + 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"Matrix the of this linear transformation is $A$(= ",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/null_space.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/null_space.py new file mode 100644 index 0000000..dfc3cb4 --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/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 $2D$ vector space(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 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 belong 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(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/solution.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/solution.py new file mode 100644 index 0000000..fb31881 --- /dev/null +++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/solution.py @@ -0,0 +1,75 @@ +from manimlib.imports import * +class solution(LinearTransformationScene): + def construct(self): + + self.setup() + self.wait() + + o = TextMobject(r"This is the original $2D$ vector space(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)) + + A = TextMobject("Let $A$ denote the matrix the of this linear transformation.") + A.move_to(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 i.e. a line ($1D$)") + o.move_to(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(DARK_BLUE) + arrow2.scale(1.2) + self.play(ShowCreation(arrow2)) + self.wait() + + o1 = TextMobject("If the vector lies in the transformed vector space") + o2 = TextMobject("(the line) then the solution exist") + o1.move_to(2*DOWN+2*RIGHT) + o2.move_to(2.5*DOWN+2*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(DARK_BLUE) + arrow1.scale(1.3) + self.play(ShowCreation(arrow1)) + self.wait() + + o1 = TextMobject("If the vector does lies in the transformed") + o2 = TextMobject("vector space then the does not solution exist") + o1.move_to(2*DOWN+2*RIGHT) + o2.move_to(2.5*DOWN+2*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 |