Merge pull request #49 from architsangal/master

Code for Orthonormal basis added

2 files changed, 152 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/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