Merge pull request #49 from architsangal/master

Code for Orthonormal basis added

3 files changed, 286 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..dd4b8d4
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file1_projections.py
@@ -0,0 +1,79 @@
+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"Consider 2 linearly independent vectors $a$ and $b$")
+        text.set_color(DARK_BLUE)
+        text.scale(0.6)
+        text.move_to(3*YTD*UP+5*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(3*UP+4*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))
\ 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/Gram Schmidt Orthonormalization Process/file_introduction.py b/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file_introduction.py
new file mode 100644
index 0000000..ccd23c9
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file_introduction.py
@@ -0,0 +1,33 @@
+from manimlib.imports import *
+
+class Orthonormal(Scene):
+    def construct(self):
+        Centre = DOWN
+        arrow_1 = Arrow(start = Centre+ORIGIN,end = Centre+1.414*(UP+RIGHT))
+        arrow_2 = Arrow(start = Centre+ORIGIN,end = Centre+2*UP)
+        arrow_1.scale(1.35)
+        arrow_2.scale(1.35)
+        text = TextMobject("This is a set of linearly independent vectors")
+        text.scale(0.75)
+        text.move_to(3*UP+3*LEFT)
+        text.set_color(PURPLE_E)
+        arrow_1.set_color(PURPLE_E)
+        arrow_2.set_color(PURPLE_E)
+        self.play(Write(text))
+        self.play(ShowCreation(arrow_1), ShowCreation(arrow_2))
+        self.wait(2)
+        text1 = TextMobject("After we apply Gram-Schmidt Orthogonalization Process to set of linearly independent vectors")
+        text1.scale(0.6)
+        text1.move_to(3*UP+2*LEFT)
+        text1.set_color(GREEN)
+        arrow_a = Arrow(start = Centre+ORIGIN,end = Centre+0.707*(UP+RIGHT))
+        arrow_a.set_color(GREEN)
+        arrow_a.scale(2)
+        self.play(Transform(text,text1))
+        self.wait(2)
+        self.play(Transform(arrow_1,arrow_a))
+        arrow_b = Arrow(start = Centre+ORIGIN,end = Centre+0.707*(UP+LEFT))
+        arrow_b.set_color(GREEN)
+        arrow_b.scale(2)
+        self.play(Transform(arrow_2,arrow_b))
+        self.wait(2)
\ No newline at end of file