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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))
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