Review

3 files changed, 336 insertions, 0 deletions

diff --git a/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file1_projections.py b/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file1_projections.py
new file mode 100755
index 0000000..814fa57
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file1_projections.py
@@ -0,0 +1,85 @@
+from manimlib.imports import *
+
+class Projections(GraphScene):
+    CONFIG = {
+        "x_min": -6,
+        "x_max": 6,
+        "y_min": -4,
+        "y_max": 4,
+        "graph_origin" : ORIGIN ,
+    }    
+    def construct(self):
+       
+        self.setup_axes(animate=True)
+
+        XTD = self.x_axis_width/(self.x_max-self.x_min)
+        YTD = self.y_axis_height/(self.y_max-self.y_min)
+
+        arrow_a = Arrow(start = ORIGIN, end = 4*XTD*RIGHT)
+        arrow_a.scale(1.2)
+        arrow_a.set_color(DARK_BLUE)
+        arrow_b = Arrow(start = ORIGIN, end = 2*YTD*UP+2*XTD*RIGHT)
+        arrow_b.scale(1.3)
+        arrow_b.set_color(DARK_BLUE)
+        self.play(ShowCreation(arrow_a), ShowCreation(arrow_b))
+
+        text = TextMobject(r"Let there be 2 vectors $a$ and $b$")
+        text.set_color(DARK_BLUE)
+        text.scale(0.75)
+        text.move_to(2*YTD*DOWN+4*XTD*LEFT)
+        text_a = TextMobject("a")
+        text_a.move_to(0.4*YTD*DOWN+3*XTD*RIGHT)
+        text_a.set_color(DARK_BLUE)
+        text_b = TextMobject("b")
+        text_b.move_to(1.5*YTD*UP+RIGHT*XTD)
+        text_b.set_color(DARK_BLUE)
+ 
+        self.play(Write(text),Write(text_a), Write(text_b))
+        self.wait()
+
+        arrow_b_copy = Arrow(start = ORIGIN, end = 2*YTD*UP+2*XTD*RIGHT)
+        arrow_b_copy.scale(1.25)
+
+        arrow_p = Arrow(start = ORIGIN, end = 2*XTD*RIGHT)
+        arrow_p.scale(1.5)
+        arrow_p.set_color(GOLD_E)
+
+        text_p = TextMobject("p")
+        text_p.move_to(0.25*DOWN+RIGHT)
+        text_p.set_color(GOLD_E)
+
+        self.play(FadeOut(text), Transform(arrow_b_copy,arrow_p), FadeOut(text_a), FadeOut(text_b))
+        text = TextMobject(r"$p$ is the projection of $b$ on $a$")
+        text.set_color(GOLD_E)
+        text.move_to(2*DOWN+3*LEFT)
+        text.scale(0.8)
+        self.play(Write(text),Write(text_p))
+        self.wait()
+        
+        self.play(FadeIn(text_a), FadeIn(text_b))
+
+        arrow_o = Arrow(start = 2*XTD*RIGHT, end = 2*YTD*UP+2*XTD*RIGHT)
+        arrow_o.scale(1.5)
+        arrow_o.set_color(GREEN_E)
+
+        text_o = TextMobject("b-p")
+        text_o.move_to(UP*YTD+2.7*XTD*RIGHT)
+        text_o.set_color(GREEN_E)
+
+        self.play(ShowCreation(arrow_o))
+        self.play(FadeOut(text),Write(text_o))
+        
+        text = TextMobject(r"Observe, ($b-p$) is orthogonal to $a$")
+        text.set_color(GREEN_E)
+        text.move_to(2*DOWN+4*LEFT)
+        text.scale(0.8)
+        self.play(Write(text))
+        self.wait(2)
+
+        self.play(FadeOut(self.axes), FadeOut(arrow_a), FadeOut(arrow_b), FadeOut(arrow_b_copy), FadeOut(arrow_o), FadeOut(text_a), FadeOut(text_b), FadeOut(text_o), FadeOut(text_p), FadeOut(text))
+        
+        text = TextMobject(r"Therefore, unit vectors of $b-p$ and $a$ are orthonormal to each other")
+        text.scale(0.75)
+        self.play(Write(text))
+        self.wait(2)
+        self.play(FadeOut(text))
\ No newline at end of file
diff --git a/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file2_orthonormal.py b/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file2_orthonormal.py
new file mode 100644
index 0000000..640b8be
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file2_orthonormal.py
@@ -0,0 +1,174 @@
+from manimlib.imports import *
+
+from manimlib.imports import *
+
+class Orthogonal(ThreeDScene):
+    def construct(self):
+
+        text = TextMobject(r"These are two Orthogonal Basis $\alpha_{1}$ and $\alpha_{2}$")
+        text.scale(0.75)
+        self.add_fixed_in_frame_mobjects(text)
+        text.move_to(3*(DOWN+RIGHT))
+        self.play(Write(text))
+
+        axes = ThreeDAxes()
+        self.play(ShowCreation(axes))
+        self.move_camera(phi=60*DEGREES,theta=45*DEGREES,run_time=3)
+        self.begin_ambient_camera_rotation(rate=0.3)
+        
+        line1 = Line(start = ORIGIN,end = 1*RIGHT)
+        line1.set_color(DARK_BLUE)
+        tip1 = Polygon(RIGHT,0.8*RIGHT-0.2*DOWN,0.8*RIGHT-0.2*UP)
+        tip1.set_opacity(1)
+        tip1.set_fill(DARK_BLUE)
+        tip1.set_color(DARK_BLUE)
+
+        arrow2 = Line(start = ORIGIN,end = 1*UP)
+        arrow2.set_color(DARK_BLUE)
+        tip2 = Polygon(UP,0.8*UP-0.2*RIGHT,0.8*UP-0.2*LEFT)
+        tip2.set_opacity(1)
+        tip2.set_fill(DARK_BLUE)
+        tip2.set_color(DARK_BLUE)
+        arrow2.set_color(DARK_BLUE)
+
+        self.play(ShowCreation(line1), ShowCreation(tip1), ShowCreation(arrow2), ShowCreation(tip2))
+        self.wait()
+
+        a_line = Line(start = ORIGIN,end = 2*UP+2*RIGHT+[0,0,2])
+        a_line.set_color(GOLD_E)
+        a_tip = Polygon(2*UP+2*RIGHT+[0,0,2],2*UP+1.6*RIGHT+[0,0,1.8],1.6*UP+2*RIGHT+[0,0,1.8])
+        a_tip.set_opacity(1)
+        a_tip.set_fill(GOLD_E)
+        a_tip.set_color(GOLD_E)
+
+        a_line_c1 = Line(start = ORIGIN,end = 2*UP+2*RIGHT+[0,0,2])
+        a_line_c1.set_color(GOLD_E)
+        a_tip_c1 = Polygon(2*UP+2*RIGHT+[0,0,2],2*UP+1.6*RIGHT+[0,0,1.8],1.6*UP+2*RIGHT+[0,0,1.8])
+        a_tip_c1.set_opacity(1)
+        a_tip_c1.set_fill(GOLD_E)
+        a_tip_c1.set_color(GOLD_E)
+
+        self.play(FadeOut(text), ShowCreation(a_line), ShowCreation(a_tip), ShowCreation(a_line_c1), ShowCreation(a_tip_c1))
+        
+        text = TextMobject(r"There is a vector $\beta_3$")
+        text.scale(0.75)
+        self.add_fixed_in_frame_mobjects(text)
+        text.move_to(3*(DOWN+RIGHT))
+        self.play(Write(text))
+        self.wait()
+        self.play(FadeOut(text))
+
+        p_line1 = Line(start = ORIGIN,end = 2*RIGHT)
+        p_line1.set_color(GOLD_E)
+        p_tip1 = Polygon(RIGHT,0.8*RIGHT+0.2*DOWN,0.8*RIGHT+0.2*UP)
+        p_tip1.move_to(2*RIGHT)
+        p_tip1.set_opacity(1)
+        p_tip1.set_fill(GOLD_E)
+        p_tip1.set_color(GOLD_E)
+
+        self.play(Transform(a_line_c1,p_line1),Transform(a_tip_c1,p_tip1))
+
+        text = TextMobject(r"Take projection of $\beta_3$ on $\alpha_1$")
+        text.scale(0.75)
+        self.add_fixed_in_frame_mobjects(text)
+        text.move_to(3*(DOWN+RIGHT))
+        self.play(Write(text))
+        self.wait()
+        self.play(FadeOut(text))
+
+        o_line1 = Line(start = 2*RIGHT,end = 2*UP+2*RIGHT+[0,0,2])
+        o_line1.set_color(GREEN_E)
+        o_tip1 = Polygon(2*UP+2*RIGHT+[0,0,2],1.8*UP+2*RIGHT+[0,0,1.8]+0.2*RIGHT,1.8*UP+2*RIGHT+[0,0,1.8]-0.2*RIGHT)
+        o_tip1.set_opacity(1)
+        o_tip1.set_fill(GREEN_E)
+        o_tip1.set_color(GREEN_E)
+
+        a_line1 = Line(start = ORIGIN,end = 2*UP+[0,0,2])
+        a_line1.set_color(GREEN_E)
+        a_tip1 = Polygon(2*UP+[0,0,2],1.8*UP+[0,0,1.8]+0.2*RIGHT,1.8*UP+[0,0,1.8]-0.2*RIGHT)
+        a_tip1.set_opacity(1)
+        a_tip1.set_fill(GREEN_E)
+        a_tip1.set_color(GREEN_E)
+
+        a_line1_c1 = Line(start = ORIGIN,end = 2*UP+[0,0,2])
+        a_line1_c1.set_color(GREEN_E)
+        a_tip1_c1 = Polygon(2*UP+[0,0,2],1.8*UP+[0,0,1.8]+0.2*RIGHT,1.8*UP+[0,0,1.8]-0.2*RIGHT)
+        a_tip1_c1.set_opacity(1)
+        a_tip1_c1.set_fill(GREEN_E)
+        a_tip1_c1.set_color(GREEN_E)
+
+        text = TextMobject(r"$\beta_3$-(projection of $\beta_3$ on $\alpha_1$)")
+        text.scale(0.75)
+        self.add_fixed_in_frame_mobjects(text)
+        text.move_to(3*(DOWN+RIGHT))
+        self.play(Write(text))
+        self.play(ShowCreation(o_line1), ShowCreation(o_tip1))
+        self.wait(2)
+        self.play(FadeOut(a_line_c1), FadeOut(a_tip_c1), 
+        FadeOut(a_line), FadeOut(a_tip),
+        Transform(o_line1,a_line1), Transform(o_tip1,a_tip1))
+
+        self.wait()
+        self.play(FadeOut(text))
+
+        p_arrow2 = Line(start = ORIGIN,end = 2*UP)
+        p_arrow2.set_color(GOLD_E)
+        p_tip2 = Polygon(2*UP,1.8*UP+0.2*RIGHT,1.8*UP+0.2*LEFT)
+        p_tip2.set_opacity(1)
+        p_tip2.set_fill(GOLD_E)
+        p_tip2.set_color(GOLD_E)
+        p_arrow2.set_color(GOLD_E)
+
+        last_a = Line(start = 2*UP,end = [0,2,2])
+        last_a.set_color(GOLD_E)
+        last_a_tip = Polygon([0,0,2],[0,0,1.8]+0.2*RIGHT,[0,0,1.8]+0.2*LEFT)
+        last_a_tip.move_to([0,2,2])
+        last_a_tip.set_opacity(1)
+        last_a_tip.set_fill(GOLD_E)
+        last_a_tip.set_color(GOLD_E)
+
+        self.wait()
+        text = TextMobject(r"Take projection on $\alpha_2$")
+        text.scale(0.75)
+        self.add_fixed_in_frame_mobjects(text)
+        text.move_to(3*(DOWN+RIGHT))
+        self.play(Write(text))
+        self.play(Transform(a_line1_c1,p_arrow2),Transform(a_tip1_c1,p_tip2))
+        self.wait()
+        self.play(FadeOut(text))
+
+        text = TextMobject(r"$\beta_3$-(projection of $\beta_3$ on $\alpha_1$ + projection of $\beta_3$ on $\alpha_2$)")
+        text.scale(0.75)
+        self.add_fixed_in_frame_mobjects(text)
+        text.move_to(3*DOWN+2*RIGHT)
+        self.play(Write(text))
+        self.play(ShowCreation(o_line1), ShowCreation(o_tip1))
+        self.wait(2)
+        self.play(ShowCreation(last_a_tip), ShowCreation(last_a))
+        self.wait()
+        self.play(FadeOut(text))
+
+        larrow3 = Line(start = ORIGIN,end = [0,0,2])
+        larrow3.set_color(GOLD_E)
+        ltip3 = Polygon([0,0,2],[0,0,1.8]+0.2*RIGHT,[0,0,1.8]+0.2*LEFT)
+        ltip3.set_opacity(1)
+        ltip3.set_fill(GOLD_E)
+        ltip3.set_color(GOLD_E)
+        self.wait()
+        self.play(FadeOut(o_line1), FadeOut(o_tip1), FadeOut(a_line1_c1), FadeOut(a_tip1_c1), Transform(last_a,larrow3), Transform(last_a_tip,ltip3))
+        
+        text = TextMobject(r"Normalisation of the orthogonal vector")
+        text.scale(0.75)
+        self.add_fixed_in_frame_mobjects(text)
+        text.move_to(3*DOWN+2*RIGHT)
+        self.play(Write(text))
+        
+        arrow3 = Line(start = ORIGIN,end = [0,0,1])
+        arrow3.set_color(DARK_BLUE)
+        tip3 = Polygon([0,0,1],[0,0,0.8]-0.2*RIGHT,[0,0,0.8]-0.2*LEFT)
+        tip3.set_opacity(1)
+        tip3.set_fill(DARK_BLUE)
+        tip3.set_color(DARK_BLUE)
+        self.play(Transform(last_a,arrow3), Transform(last_a_tip,tip3))
+
+        self.wait(3)
\ No newline at end of file
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