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-rw-r--r--FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/README.md15
-rwxr-xr-xFSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file1_orthogonal.py40
-rwxr-xr-xFSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file2_sum_of_projections_part1.py133
-rw-r--r--FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file3_sum_of_projections_part2.py181
-rw-r--r--FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file4.gifbin0 -> 1733367 bytes
-rw-r--r--FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file5.gifbin0 -> 5430037 bytes
-rw-r--r--FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file6.gifbin0 -> 21739496 bytes
7 files changed, 369 insertions, 0 deletions
diff --git a/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/README.md b/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/README.md
new file mode 100644
index 0000000..e287fa1
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/README.md
@@ -0,0 +1,15 @@
+# Contributer: Archit Sangal
+My Github Account : <a href="https://github.com/architsangal">architsangal</a> (https://github.com/architsangal)
+<br/></br>
+
+## Sub-Topics Covered:
++ Orthonormal Bases
+
+#### Video 1: Example of Orthonormal bases
+![GIF1](file4.gif)
+
+#### Video 2: Adding the projections of a vector on orthonormal basis will produce the same vector
+![GIF2](file5.gif)
+
+#### Video 3: Relating the example and the property
+![GIF3](file6.gif) \ No newline at end of file
diff --git a/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file1_orthogonal.py b/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file1_orthogonal.py
new file mode 100755
index 0000000..a5d96f5
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file1_orthogonal.py
@@ -0,0 +1,40 @@
+from manimlib.imports import *
+
+class Orthogonal(ThreeDScene):
+ def construct(self):
+ axes = ThreeDAxes()
+ self.play(ShowCreation(axes))
+ self.move_camera(phi=60*DEGREES,theta=45*DEGREES,run_time=3)
+
+ text = TextMobject(r"$\hat{i}$",r"$\hat{j}$",r"$\hat{k}$")
+ text[0].move_to(0.7*DOWN+0.8*LEFT)
+ text[1].move_to(0.75*DOWN+0.7*RIGHT)
+ text[2].move_to(0.75*UP+0.4*RIGHT)
+ self.add_fixed_in_frame_mobjects(text)
+ self.play(Write(text))
+
+ line1 = Line(start = ORIGIN,end = RIGHT)
+ line1.set_color(DARK_BLUE)
+ tip1 = Polygon(-0.95*LEFT,-0.8*LEFT-0.1*DOWN,-0.8*LEFT-0.1*UP)
+ tip1.set_opacity(1)
+ tip1.set_fill(DARK_BLUE)
+ tip1.set_color(DARK_BLUE)
+
+ arrow2 = Line(start = ORIGIN,end = UP)
+ arrow2.set_color(DARK_BLUE)
+ tip2 = Polygon(0.95*UP,0.8*UP-0.1*RIGHT,0.8*UP-0.1*LEFT)
+ tip2.set_opacity(1)
+ tip2.set_fill(DARK_BLUE)
+ tip2.set_color(DARK_BLUE)
+ arrow2.set_color(DARK_BLUE)
+
+ arrow3 = Line(start = ORIGIN,end = [0,0,1])
+ arrow3.set_color(DARK_BLUE)
+ tip3 = Polygon([0,0,0.95],[0,0,0.8]-0.1*RIGHT,[0,0,0.8]-0.1*LEFT)
+ tip3.set_opacity(1)
+ tip3.set_fill(DARK_BLUE)
+ tip3.set_color(DARK_BLUE)
+
+ self.play(ShowCreation(line1), ShowCreation(tip1), ShowCreation(arrow2), ShowCreation(tip2), ShowCreation(arrow3), ShowCreation(tip3))
+
+ self.wait()
diff --git a/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file2_sum_of_projections_part1.py b/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file2_sum_of_projections_part1.py
new file mode 100755
index 0000000..141e99b
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file2_sum_of_projections_part1.py
@@ -0,0 +1,133 @@
+from manimlib.imports import *
+class LinearTrans(LinearTransformationScene):
+ CONFIG = {
+ "show_basis_vectors": True,
+ "basis_vector_stroke_width": 1,
+ "leave_ghost_vectors": False,
+ "show_coordinates": True,
+ }
+
+ def construct(self):
+
+ self.setup()
+
+ matrix = [[0.6,-0.8],[0.8,0.6]]
+ self.apply_matrix(matrix)
+
+ self.wait(2)
+ orthonormal = TextMobject(r"These are 2 orthonormal vectors($v_1$ and $v_2$)")
+ orthonormal.scale(0.7)
+ orthonormal.move_to(DOWN+LEFT*3.5)
+ orthonormal.add_background_rectangle()
+ v1 = TextMobject(r"$v_1$")
+ v1.scale(0.75)
+ v1.set_color(X_COLOR)
+ v1.move_to(0.75*UP+RIGHT)
+ v1.add_background_rectangle()
+ v2 = TextMobject(r"$v_2$")
+ v2.scale(0.75)
+ v2.set_color(Y_COLOR)
+ v2.move_to(0.75*UP+LEFT*1.25)
+ v2.add_background_rectangle()
+ self.play(Write(orthonormal))
+ self.play(Write(v1),Write(v2))
+ self.wait()
+ self.play(FadeOut(orthonormal), FadeOut(v1), FadeOut(v2))
+
+ arrow = Arrow(start = ORIGIN,end = 3*RIGHT+UP)
+ arrow.scale(1.2)
+ arrow.set_color(BLUE)
+ arrow.apply_matrix(matrix)
+ text3 = TextMobject("v")
+ text3.move_to(3.2*UP+1.2*RIGHT)
+ text3.add_background_rectangle()
+ self.play(ShowCreation(arrow),Write(text3))
+ self.wait()
+ v_cor = TextMobject("(1 , 3)")
+ v_cor.move_to(3.2*UP+1.3*RIGHT)
+ v_cor.set_color(BLUE)
+ v_cor.scale(0.75)
+ v_cor.add_background_rectangle()
+ self.play(Transform(text3,v_cor))
+
+ line1 = DashedLine(start = 1*UP+3*RIGHT, end = 3*RIGHT)
+ line2 = DashedLine(start = 1*UP+3*RIGHT, end = UP)
+ line1.apply_matrix(matrix)
+ line2.apply_matrix(matrix)
+ self.play(ShowCreation(line1),ShowCreation(line2),run_time = 2)
+
+ v1 = Arrow(start = ORIGIN,end = 3*RIGHT+UP)
+ v1.scale(1.2)
+ v1.set_color(BLUE)
+ v1.apply_matrix(matrix)
+ arrow1 = Arrow(start = ORIGIN,end = 3*RIGHT)
+ arrow1.scale(1.2)
+ arrow1.set_color("#6B8E23")
+ arrow1.apply_matrix(matrix)
+ self.play(Transform(v1,arrow1))
+ v1_cor = TextMobject(r"$<v,v_1> v_1$")
+ v1_cor.move_to(2.5*UP+3*RIGHT)
+ v1_cor.scale(0.75)
+ v1_cor.add_background_rectangle()
+ self.play(Write(v1_cor))
+ self.wait(0.5)
+ text1 = TextMobject("(1.8 , 2.4)")
+ text1.move_to(2.1*UP+2.5*RIGHT)
+ text1.scale(0.75)
+ text1.set_color("#6B8E23")
+ text1.add_background_rectangle()
+ self.play(Transform(v1_cor,text1))
+
+ v2 = Arrow(start = ORIGIN,end = 3*RIGHT+UP)
+ v2.scale(1.2)
+ v2.set_color("#8b0000")
+ v2.apply_matrix(matrix)
+ arrow2 = Arrow(start = ORIGIN,end = UP)
+ arrow2.scale(2.1)
+ arrow2.set_color("#8b0000")
+ arrow2.apply_matrix(matrix)
+ self.wait(0.5)
+ self.play(Transform(v2,arrow2))
+ self.wait(0.5)
+ v2_cor = TextMobject(r"$<v,v_2> v_2$")
+ v2_cor.move_to(0.75*UP+2.5*LEFT)
+ v2_cor.scale(0.75)
+ v2_cor.add_background_rectangle()
+ self.play(Write(v2_cor))
+ self.wait(0.5)
+ text2 = TextMobject("(-0.8 , 0.6)")
+ text2.move_to(0.75*UP+1.75*LEFT)
+ text2.scale(0.75)
+ text2.set_color("#8b0000")
+ text2.add_background_rectangle()
+ self.play(Transform(v2_cor,text2))
+
+ self.wait()
+
+ self.play(ApplyMethod(v2.move_to,1.4*RIGHT+2.7*UP),FadeOut(v1_cor),FadeOut(v2_cor),FadeOut(v_cor))
+
+ self.wait()
+
+ ending = TextMobject(r"$v = <v,v_1> v_1 + <v,v_2> v_2$")
+ ending.scale(0.7)
+ ending.move_to(DOWN)
+ ending.add_background_rectangle()
+ self.play(Write(ending))
+ self.wait()
+ self.play(FadeOut(ending))
+
+ ending = TextMobject(r"$\left[ \begin{array} {c} 1\\ 3 \end{array}\right] = \left[ \begin{array} {c}1.8 \\ 2.4 \end{array}\right] + \left[ \begin{array} {c} -0.8\\ 0.6 \end{array}\right]$")
+ ending.scale(0.7)
+ ending.move_to(DOWN)
+ ending.add_background_rectangle()
+ self.play(Write(ending))
+ self.wait()
+ self.play(FadeOut(ending))
+
+ ending = TextMobject(r"$v$ is the sum of projections on the orthonormal vectors")
+ ending.scale(0.7)
+ ending.move_to(DOWN)
+ ending.add_background_rectangle()
+ self.play(Write(ending))
+ self.wait()
+ self.play(FadeOut(ending)) \ No newline at end of file
diff --git a/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file3_sum_of_projections_part2.py b/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file3_sum_of_projections_part2.py
new file mode 100644
index 0000000..2899286
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file3_sum_of_projections_part2.py
@@ -0,0 +1,181 @@
+from manimlib.imports import *
+class ThreeDExplanation(ThreeDScene):
+
+ def construct(self):
+
+ basis = TextMobject(r"Set of Orthonormal Basis - $\left(\begin{array}{c}\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\\0\end{array}\right),\left(\begin{array}{c}\frac{-1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\\0\end{array}\right),\left(\begin{array}{c}0\\0\\1\end{array}\right)$")
+ basis.scale(0.75)
+ basis.move_to(UP*1.5)
+ self.play(Write(basis))
+ v = TextMobject(r"$v_1$",r"$v_2$",r"$v_3$")
+ v[0].move_to(UP*0.5+RIGHT*0.75)
+ v[1].move_to(UP*0.5+RIGHT*2.5)
+ v[2].move_to(UP*0.5+RIGHT*4)
+ eq = TextMobject(r"$v = \left(\begin{array}{c}3\\4\\5\end{array}\right)$")
+ eq1 = TextMobject(r"$<v,v_1> = \frac{3}{\sqrt{2}} + \frac{4}{\sqrt{2}} + 0 = \frac{7}{\sqrt{2}}$")
+ eq2 = TextMobject(r"$<v,v_2> = \frac{-3}{\sqrt{2}} + \frac{4}{\sqrt{2}} + 0 =\frac{1}{\sqrt{2}}$")
+ eq3 = TextMobject(r"$<v,v_3> =  0 + 0 + 5 =5$")
+ eq.move_to(4*LEFT+DOWN)
+ eq1.move_to(0.5*DOWN+2*RIGHT)
+ eq2.move_to(1.5*DOWN+2*RIGHT)
+ eq3.move_to(2.5*DOWN+2*RIGHT)
+ self.play(Write(v))
+ self.play(Write(eq))
+ self.play(Write(eq1))
+ self.play(Write(eq2))
+ self.play(Write(eq3))
+ self.wait()
+ self.play(FadeOut(basis), FadeOut(eq), FadeOut(v), FadeOut(eq1), FadeOut(eq2), FadeOut(eq3))
+ self.wait()
+
+ text = TextMobject("These are the 3 mutually orthonormal basis of the set(", r"$v_1$, ", r"$v_2$, ", r"$v_3$",")")
+ text[1].set_color(DARK_BLUE)
+ text[2].set_color(RED)
+ text[3].set_color(YELLOW)
+ text.scale(0.75)
+ self.add_fixed_in_frame_mobjects(text)
+ text.move_to(3*DOWN)
+ self.play(Write(text))
+ self.wait()
+
+ axes = ThreeDAxes(x_min = -6,x_max=6,y_min=-6,y_max=6,z_min=-6,z_max=6)
+ self.play(ShowCreation(axes))
+ self.move_camera(distance = 100, phi=45*DEGREES,theta=45*DEGREES,run_time=5)
+ self.begin_ambient_camera_rotation(rate=0.1)
+
+ xy_plane = Polygon(6*RIGHT+6*UP,-6*RIGHT+6*UP,-6*RIGHT-6*UP,6*RIGHT-6*UP)
+ xy_plane.set_color("#333333")
+ xy_plane.set_fill("#333333")
+ xy_plane.set_opacity(1)
+ xy_plane.fade(0.7)
+ self.play(ShowCreation(xy_plane))
+
+ dashedline1 = DashedLine(start = -6*(UP+RIGHT), end = 6*(UP+RIGHT))
+ dashedline2 = DashedLine(start = -6*(UP+LEFT), end = 6*(UP+LEFT))
+ dashedline3 = DashedLine(start = 4*UP+3*RIGHT+[0,0,5], end = 3.5*UP+3.5*RIGHT)
+ dashedline4 = DashedLine(start = 4*UP+3*RIGHT+[0,0,5], end = 0.5*UP+0.5*LEFT)
+ dashedline5 = DashedLine(start = 4*UP+3*RIGHT+[0,0,5], end = [0,0,5])
+
+ self.play(ShowCreation(dashedline1), ShowCreation(dashedline2))
+
+ line1 = Line(start = ORIGIN,end = 0.707*RIGHT + 0.707*UP)
+ line1.set_color(DARK_BLUE)
+ tip1 = Polygon(0.707*RIGHT + 0.707*UP, 0.707*RIGHT + 0.607*UP, 0.607*RIGHT + 0.707*UP)
+ tip1.set_opacity(1)
+ tip1.set_fill(DARK_BLUE)
+ tip1.set_color(DARK_BLUE)
+ self.play(ShowCreation(line1), ShowCreation(tip1))
+
+ line2 = Line(start = ORIGIN,end = 0.707*LEFT + 0.707*UP)
+ line2.set_color(RED)
+ tip2 = Polygon(0.707*LEFT + 0.707*UP, 0.707*LEFT + 0.607*UP, 0.607*LEFT + 0.707*UP)
+ tip2.set_opacity(1)
+ tip2.set_fill(RED)
+ tip2.set_color(RED)
+
+ self.play(ShowCreation(line2), ShowCreation(tip2))
+
+ line3 = Line(start = ORIGIN,end = [0,0,1])
+ line3.set_color(YELLOW)
+ 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(YELLOW)
+ tip3.set_color(YELLOW)
+ self.play(ShowCreation(line3), ShowCreation(tip3))
+ self.wait()
+
+ self.play(FadeOut(text))
+
+ text = TextMobject("Take the projection of ", r"$v$", " on the mutually orthonormal vectors")
+ text[1].set_color(GOLD_E)
+ text.scale(0.75)
+ self.add_fixed_in_frame_mobjects(text)
+ text.move_to(3*DOWN)
+ self.play(Write(text))
+ self.wait(2)
+
+ a_line = Line(start = ORIGIN,end = 4*UP+3*RIGHT+[0,0,5])
+ a_line.set_color(GOLD_E)
+ a_tip = Polygon(3.92*UP+2.94*RIGHT+[0,0,4.9],3.6*UP+2.7*RIGHT+[0,0,4.5]+0.1*UP+0.1*LEFT,3.6*UP+2.7*RIGHT+[0,0,4.5]+0.1*DOWN+0.1*RIGHT)
+ a_tip.set_opacity(1)
+ a_tip.set_fill(GOLD_E)
+ a_tip.set_color(GOLD_E)
+
+ self.play(ShowCreation(a_line), ShowCreation(a_tip))
+ self.stop_ambient_camera_rotation()
+ self.move_camera(distance = 100, phi=45*DEGREES,theta=135*DEGREES,run_time=5)
+
+ self.play(ShowCreation(dashedline3),ShowCreation(dashedline4),ShowCreation(dashedline5))
+ self.wait()
+
+ pv1 = Line(start = ORIGIN,end = 4*UP+3*RIGHT+[0,0,5])
+ pv1.set_color(GOLD_E)
+ pv1tip = Polygon(4*UP+3*RIGHT+[0,0,5],3.6*UP+2.7*RIGHT+[0,0,4.5]+0.1*UP+0.1*LEFT,3.6*UP+2.7*RIGHT+[0,0,4.5]+0.1*DOWN+0.1*RIGHT)
+ pv1tip.set_opacity(1)
+ pv1tip.set_fill(GOLD_E)
+ pv1tip.set_color(GOLD_E)
+
+ v1_p = Line(start = ORIGIN,end = 3.5*RIGHT + 3.5*UP)
+ v1_p.set_color(BLUE_E)
+ v1_p_tip = Polygon(3.5*RIGHT + 3.5*UP, 3.5*RIGHT + 3.4*UP, 3.4*RIGHT + 3.5*UP)
+ v1_p_tip.set_opacity(1)
+ v1_p_tip.set_fill(BLUE_E)
+ v1_p_tip.set_color(BLUE_E)
+
+ pv2 = Line(start = ORIGIN,end = 4*UP+3*RIGHT+[0,0,5])
+ pv2.set_color(GOLD_E)
+ pv2tip = Polygon(4*UP+3*RIGHT+[0,0,5],3.6*UP+2.7*RIGHT+[0,0,4.5]+0.1*UP+0.1*LEFT,3.6*UP+2.7*RIGHT+[0,0,4.5]+0.1*DOWN+0.1*RIGHT)
+ pv2tip.set_opacity(1)
+ pv2tip.set_fill(GOLD_E)
+ pv2tip.set_color(GOLD_E)
+
+ v2_p = Line(start = ORIGIN,end = 0.5*LEFT + 0.5*UP)
+ v2_p.set_color(RED_E)
+ v2_p_tip = Polygon(0.5*LEFT + 0.5*UP, 0.5*LEFT + 0.4*UP, 0.4*LEFT + 0.5*UP)
+ v2_p_tip.set_opacity(1)
+ v2_p_tip.set_fill(RED_E)
+ v2_p_tip.set_color(RED_E)
+
+ pv3 = Line(start = ORIGIN,end = 4*UP+3*RIGHT+[0,0,5])
+ pv3.set_color(GOLD_E)
+ pv3tip = Polygon(4*UP+3*RIGHT+[0,0,5],3.6*UP+2.7*RIGHT+[0,0,4.5]+0.1*UP+0.1*LEFT,3.6*UP+2.7*RIGHT+[0,0,4.5]+0.1*DOWN+0.1*RIGHT)
+ pv3tip.set_opacity(1)
+ pv3tip.set_fill(GOLD_E)
+ pv3tip.set_color(GOLD_E)
+
+ v3_p = Line(start = ORIGIN,end = [0,0,5])
+ v3_p.set_color(YELLOW_E)
+ v3_p_tip = Polygon([0,0,5.15],[0,0,4.8]+0.2*RIGHT,[0,0,4.8]+0.2*LEFT)
+ v3_p_tip.set_opacity(1)
+ v3_p_tip.set_fill(YELLOW_E)
+ v3_p_tip.set_color(YELLOW_E)
+
+ #self.stop_ambient_camera_rotation()
+ self.play(Transform(pv1,v1_p),
+ Transform(pv1tip,v1_p_tip),
+ Transform(pv2,v2_p),
+ Transform(pv2tip,v2_p_tip),
+ Transform(pv3,v3_p),
+ Transform(pv3tip,v3_p_tip))
+ self.play(FadeOut(dashedline1),
+ FadeOut(dashedline2),
+ FadeOut(dashedline3),
+ FadeOut(dashedline4),
+ FadeOut(dashedline5),
+ FadeOut(line1),
+ FadeOut(tip1),
+ FadeOut(line2),
+ FadeOut(tip2),
+ FadeOut(line3),
+ FadeOut(tip3),
+ FadeOut(text))
+
+ text = TextMobject(r"$v$ is the sum of projections on the orthonormal vectors")
+ text.set_color(GOLD_E)
+ text.scale(0.75)
+ self.add_fixed_in_frame_mobjects(text)
+ text.move_to(3*DOWN)
+ self.play(Write(text), ApplyMethod(pv2.move_to,(3.5*RIGHT + 3.5*UP+3*RIGHT+4*UP)/2), ApplyMethod(pv2tip.move_to,(3.1*RIGHT + 3.9*UP)))
+ self.play(ApplyMethod(pv3.move_to,3*RIGHT + 4*UP + [0,0,2.5]), ApplyMethod(pv3tip.move_to,(3*RIGHT + 4*UP + [0,0,4.8])))
+
+ self.wait(3) \ No newline at end of file
diff --git a/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file4.gif b/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file4.gif
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+++ b/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file4.gif
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diff --git a/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file5.gif b/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file5.gif
new file mode 100644
index 0000000..d7eb0bc
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+++ b/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file5.gif
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diff --git a/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file6.gif b/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file6.gif
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+++ b/FSF-2020/linear-algebra/linear-transformations/Orthonormal-Basis/file6.gif
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generated by cgit (git 2.34.1) at 2024-12-20 18:30:27 +0000