Merge pull request #50 from architsangal/master

Orthonormal Basis, Gram-Schmidt Orthogonalization Process and Linear Transformation

3 files changed, 294 insertions, 49 deletions

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
index 640b8be..af51fc6 100644
--- 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
@@ -1,18 +1,153 @@
 from manimlib.imports import *
 
-from manimlib.imports import *
-
-class Orthogonal(ThreeDScene):
+class Algo(ThreeDScene):
     def construct(self):
 
-        text = TextMobject(r"These are two Orthogonal Basis $\alpha_{1}$ and $\alpha_{2}$")
+        axes = ThreeDAxes(x_min = -5,x_max=5,y_min=-3,y_max=3,z_min=-4,z_max=4)
+        self.play(ShowCreation(axes))
+
+        text = TextMobject(r"This is the vector $\beta_1(=\left[\begin{array}{c} 4\\0\\0 \end{array}\right])$")
+        text.set_color(GREEN)
+        text.scale(0.6)
+        text.move_to(3*UP+5*LEFT)
+        self.play(Write(text))
+
+        arrow_a = Arrow(start = ORIGIN, end = 4*RIGHT)
+        arrow_a.set_color(GREEN)
+        arrow_a.scale(1.15)
+        self.play(ShowCreation(arrow_a))
+
+        text_a = TextMobject(r"$\beta_1$")
+        text_a.move_to(0.4*DOWN+3*RIGHT)
+        text_a.set_color(GREEN)
+        text_a.scale(0.75)
+        self.play(Write(text_a))
+        self.wait()
+        self.play(FadeOut(text))
+
+        text = TextMobject(r"Normalize $\beta_1$ to get $\alpha_1$")
+        text.set_color(DARK_BLUE)
         text.scale(0.75)
+        text.move_to(3*UP+5*LEFT)
+        self.play(Write(text))
+
+        alpha_1 = Arrow(start = ORIGIN,end = RIGHT)
+        alpha_1.scale(1.9)
+        alpha_1.set_color(DARK_BLUE)
+        text_alpha_1 = TextMobject(r"$\alpha_1$")
+        text_alpha_1.move_to(0.4*DOWN+RIGHT)
+        text_alpha_1.set_color(DARK_BLUE)
+        text_alpha_1.scale(0.75)
+        self.play(Transform(text_a,text_alpha_1), Transform(arrow_a,alpha_1))
+        self.wait()
+        self.play(FadeOut(text))
+
+        text = TextMobject(r"Consider another vector $\beta_2(=\left[\begin{array}{c} 2\\2\\0 \end{array}\right])$")
+        text1 = TextMobject(r"which is linearly independent to $\beta_1$")
+        text.set_color(GREEN)
+        text1.set_color(GREEN)
+        text.scale(0.6)
+        text1.scale(0.6)
+        text.move_to(3*UP+4*LEFT)
+        text1.move_to(2*UP+4*LEFT)
+        self.play(Write(text))
+        self.play(Write(text1))
+
+        arrow_b = Arrow(start = ORIGIN, end = 2*UP+2*RIGHT)
+        arrow_b.scale(1.2)
+        arrow_b.set_color(GREEN)
+        text_b = TextMobject(r"$\beta_2$")
+        text_b.move_to(1.5*UP+RIGHT)
+        text_b.set_color(GREEN)
+        text_b.scale(0.75)
+ 
+        self.play(ShowCreation(arrow_b), Write(text_b))
+        self.wait()
+
+        arrow_b_copy = Arrow(start = ORIGIN, end = 2*UP+2*RIGHT)
+        arrow_b_copy.scale(1.2)
+
+        arrow_p = Arrow(start = ORIGIN, end = 2*RIGHT)
+        arrow_p.scale(1.35)
+        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), FadeOut(text1), Transform(arrow_b_copy,arrow_p), FadeOut(text_a), FadeOut(text_b))
+        text = TextMobject(r"$p$ is the projection of $\beta_2$ on $\alpha_1$")
+        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_b))
+
+        arrow_o = Arrow(start = 2*RIGHT, end = 2*UP+2*RIGHT)
+        arrow_o.scale(1.35)
+        arrow_o.set_color(PURPLE_E)
+
+        text_o = TextMobject(r"$\beta_2-p$")
+        text_o.move_to(UP+2.7*RIGHT)
+        text_o.scale(0.75)
+        text_o.set_color(PURPLE_E)
+
+        self.play(ShowCreation(arrow_o))
+        self.play(FadeOut(text),Write(text_o))
+        
+        text = TextMobject(r"$\beta_2-p$ is orthogonal to p")
+        text1 = TextMobject(r"(and hence orthogonal to $\alpha_1$ also)")
+        text.set_color(PURPLE_E)
+        text1.set_color(PURPLE_E)
+        text.scale(0.7)
+        text1.scale(0.7)
+        text.move_to(3*UP+4*LEFT)
+        text1.move_to(2.5*UP+4*LEFT)
+        self.play(Write(text))
+        self.play(Write(text1))
+        self.wait(2)
+
+        self.play(FadeOut(text_p), FadeIn(arrow_a), FadeOut(text), FadeOut(text1), FadeOut(arrow_b_copy), FadeOut(arrow_p), FadeOut(text_b), FadeOut(arrow_b))
+        self.play(ApplyMethod(arrow_o.move_to,UP), ApplyMethod(text_o.move_to,RIGHT+UP))
+
+        text = TextMobject(r"Now, Normalize $\beta_2-p$")
+        text.set_color(DARK_BLUE)
+        text.scale(0.6)
+        text.move_to(3*UP+4*LEFT)
+        self.play(Write(text))
+
+        alpha_2 = Arrow(start = ORIGIN,end = UP)
+        alpha_2.scale(1.9)
+        alpha_2.set_color(DARK_BLUE)
+        text_alpha_2 = TextMobject(r"$\alpha_2$")
+        text_alpha_2.move_to(0.4*LEFT+UP)
+        text_alpha_2.set_color(DARK_BLUE)
+        text_alpha_2.scale(0.75)
+        self.play(Transform(text_o,text_alpha_2), Transform(arrow_o,alpha_2), FadeIn(text_a))
+        self.wait()
+        self.play(FadeOut(text),FadeOut(text_a),FadeOut(text_o))
+
+        self.add(axes)
+        #############################################################################
+        axis = TextMobject(r"$\alpha_1$",r"$\alpha_2$",r"$\alpha_3$",r"$\beta_3$",r"$\alpha_3$",r"$\alpha_3$",r"$\alpha_3$",r"$\alpha_3$")
+        axis.scale(0.5)
+        axis[0].move_to(0.5*RIGHT+[0,0,-0.5])
+        axis[1].move_to(0.5*UP+[0,0,-0.5])
+        axis[2].move_to(np.array([0,0,0.5]))
+        axis[3].move_to(np.array([1,1,1.5]))
+        self.add_fixed_orientation_mobjects(axis[0])
+        self.add_fixed_orientation_mobjects(axis[1])
+        #############################################################################
+
+        text = TextMobject(r"These are the same two orthonormal vectors $\alpha_{1}$ and $\alpha_{2}$")
+        text.scale(0.6)
+        text.set_color(DARK_BLUE)
         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)
         
@@ -31,7 +166,7 @@ class Orthogonal(ThreeDScene):
         tip2.set_color(DARK_BLUE)
         arrow2.set_color(DARK_BLUE)
 
-        self.play(ShowCreation(line1), ShowCreation(tip1), ShowCreation(arrow2), ShowCreation(tip2))
+        self.play(ShowCreation(line1), ShowCreation(tip1), ShowCreation(arrow2), ShowCreation(tip2), FadeOut(arrow_a), FadeOut(arrow_o))
         self.wait()
 
         a_line = Line(start = ORIGIN,end = 2*UP+2*RIGHT+[0,0,2])
@@ -50,9 +185,11 @@ class Orthogonal(ThreeDScene):
 
         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)
+        text = TextMobject(r"Now, we have a vector $\beta_3(=\left[\begin{array}{c} 2\\2\\2 \end{array}\right])$")
+        text.set_color(GOLD_E)
+        text.scale(0.7)
         self.add_fixed_in_frame_mobjects(text)
+        self.add_fixed_orientation_mobjects(axis[3])
         text.move_to(3*(DOWN+RIGHT))
         self.play(Write(text))
         self.wait()
@@ -69,7 +206,8 @@ class Orthogonal(ThreeDScene):
         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)
+        text.scale(0.6)
+        text.set_color(GOLD_E)
         self.add_fixed_in_frame_mobjects(text)
         text.move_to(3*(DOWN+RIGHT))
         self.play(Write(text))
@@ -98,14 +236,15 @@ class Orthogonal(ThreeDScene):
         a_tip1_c1.set_color(GREEN_E)
 
         text = TextMobject(r"$\beta_3$-(projection of $\beta_3$ on $\alpha_1$)")
-        text.scale(0.75)
+        text.set_color(GREEN_E)
+        text.scale(0.6)
         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),
+        FadeOut(a_line), FadeOut(a_tip), FadeOut(axis[3]),
         Transform(o_line1,a_line1), Transform(o_tip1,a_tip1))
 
         self.wait()
@@ -120,16 +259,17 @@ class Orthogonal(ThreeDScene):
         p_arrow2.set_color(GOLD_E)
 
         last_a = Line(start = 2*UP,end = [0,2,2])
-        last_a.set_color(GOLD_E)
+        last_a.set_color(PURPLE_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)
+        last_a_tip.set_fill(PURPLE_E)
+        last_a_tip.set_color(PURPLE_E)
 
         self.wait()
         text = TextMobject(r"Take projection on $\alpha_2$")
-        text.scale(0.75)
+        text.scale(0.6)
+        text.set_color(GOLD_E)
         self.add_fixed_in_frame_mobjects(text)
         text.move_to(3*(DOWN+RIGHT))
         self.play(Write(text))
@@ -138,9 +278,10 @@ class Orthogonal(ThreeDScene):
         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)
+        text.set_color(PURPLE_E)
+        text.scale(0.6)
         self.add_fixed_in_frame_mobjects(text)
-        text.move_to(3*DOWN+2*RIGHT)
+        text.move_to(3*DOWN+3.5*RIGHT)
         self.play(Write(text))
         self.play(ShowCreation(o_line1), ShowCreation(o_tip1))
         self.wait(2)
@@ -149,19 +290,26 @@ class Orthogonal(ThreeDScene):
         self.play(FadeOut(text))
 
         larrow3 = Line(start = ORIGIN,end = [0,0,2])
-        larrow3.set_color(GOLD_E)
+        larrow3.set_color(PURPLE_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)
+        ltip3.set_fill(PURPLE_E)
+        ltip3.set_color(PURPLE_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)
+        text = TextMobject(r"Normalize, the vector")
+        text1 = TextMobject(r"$\beta_3$-(projection of $\beta_3$ on $\alpha_1$ + projection of $\beta_3$ on $\alpha_2$")
+        text.set_color(PURPLE_E)
+        text1.set_color(PURPLE_E)
+        text.scale(0.6)
+        text1.scale(0.6)
         self.add_fixed_in_frame_mobjects(text)
-        text.move_to(3*DOWN+2*RIGHT)
+        self.add_fixed_in_frame_mobjects(text1)
+        text.move_to(3*DOWN+3*RIGHT)
+        text1.move_to(3.5*DOWN+3*RIGHT)
         self.play(Write(text))
+        self.play(Write(text1))
         
         arrow3 = Line(start = ORIGIN,end = [0,0,1])
         arrow3.set_color(DARK_BLUE)
@@ -170,5 +318,16 @@ class Orthogonal(ThreeDScene):
         tip3.set_fill(DARK_BLUE)
         tip3.set_color(DARK_BLUE)
         self.play(Transform(last_a,arrow3), Transform(last_a_tip,tip3))
+        self.add_fixed_orientation_mobjects(axis[2])
+
+        self.wait()
+        self.play(FadeOut(text),FadeOut(text1))
+
+        text = TextMobject(r"These are the three orthonormal vectors $\alpha_1, \alpha_2, \alpha_3$")
+        text.set_color(DARK_BLUE)
+        self.add_fixed_in_frame_mobjects(text)
+        text.scale(0.6)
+        text.move_to(3*DOWN+3.5*RIGHT)
+        self.play(Write(text))
 
         self.wait(3)
\ No newline at end of file
diff --git a/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file3_Non_Standard_Basis.py b/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file3_Non_Standard_Basis.py
new file mode 100644
index 0000000..6410a2c
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file3_Non_Standard_Basis.py
@@ -0,0 +1,51 @@
+from manimlib.imports import *
+
+class NSB(ThreeDScene):
+    def construct(self):
+
+        axes = ThreeDAxes(x_min = -4,x_max=4,y_min=-4,y_max=4,z_min=-4,z_max=4)
+        self.play(ShowCreation(axes))
+        self.move_camera(phi=60*DEGREES,theta=45*DEGREES,run_time=3)
+        self.begin_ambient_camera_rotation(rate=0.5)
+
+        matrix = [[0.577,0.577,0.577],[-0.577,0.577,0.577],[0.577,-0.577,0.577]]
+
+        line1 = Line(start = ORIGIN,end = 1*RIGHT)
+        line1.set_color(DARK_BLUE)
+        tip1 = Polygon(RIGHT,0.9*RIGHT-0.1*DOWN,0.9*RIGHT-0.1*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,0.9*UP-0.1*RIGHT,0.9*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([0,0,1],[0,0,0.9]-0.1*RIGHT,[0,0,0.9]-0.1*LEFT)
+        tip3.set_opacity(1)
+        tip3.set_fill(DARK_BLUE)
+        tip3.set_color(DARK_BLUE)
+
+        line1.apply_matrix(matrix)
+        tip1.apply_matrix(matrix)
+        arrow2.apply_matrix(matrix)
+        tip2.apply_matrix(matrix)
+        arrow3.apply_matrix(matrix)
+        tip3.apply_matrix(matrix)
+
+        self.play(ShowCreation(line1), ShowCreation(tip1), ShowCreation(arrow2), ShowCreation(tip2), ShowCreation(arrow3), ShowCreation(tip3))
+        
+        text = TextMobject(r"This is also a set of Orthonormal Vectors")
+        text.set_color(DARK_BLUE)
+        self.add_fixed_in_frame_mobjects(text)
+        text.scale(0.6)
+        text.move_to(3*DOWN+3.5*RIGHT)
+        self.play(Write(text))
+
+        self.wait(7)
\ No newline at end of file
diff --git a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file_before_matrix.py b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file_before_matrix.py
index 85a2983..e0476a3 100644..100755
--- a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file_before_matrix.py
+++ b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file_before_matrix.py
@@ -1,13 +1,13 @@
 from manimlib.imports import *
 
-class linear(GraphScene):
+class Linear(GraphScene):
 
     CONFIG = {
     "x_min": -5,
     "x_max": 5,
     "y_min": -5,
     "y_max": 5,
-    "graph_origin": ORIGIN+RIGHT,
+    "graph_origin": ORIGIN,
     "x_labeled_nums": list(range(-5, 6)),
     "y_labeled_nums": list(range(-5, 6)),
     "x_axis_width": 7,
@@ -22,6 +22,7 @@ class linear(GraphScene):
         self.setup_axes(animate = True)
         heading = TextMobject(r"$T(x,y) = T(x+2y,x-y)$")
         heading.move_to(UP*3+LEFT*4)
+        heading.scale(0.7)
         self.play(Write(heading))
         self.wait()
 
@@ -34,44 +35,42 @@ class linear(GraphScene):
         dot1.set_color(DARK_BLUE)
         dot2.set_color(DARK_BLUE)
         p1 = TextMobject(r"$P_1$")
+        p1.scale(0.75)
         p1.set_color(DARK_BLUE)
-        p1.move_to(self.graph_origin+1*XTD*RIGHT+2*YTD*UP)
+        p1.move_to(self.graph_origin+1*XTD*RIGHT+1.5*YTD*UP)
         p2 = TextMobject(r"$P_2$")
         p2.set_color(DARK_BLUE)
-        p2.move_to(self.graph_origin+2*XTD*RIGHT+2*YTD*UP)
+        p2.scale(0.75)
+        p2.move_to(self.graph_origin+2*XTD*RIGHT+1.5*YTD*UP)
 
         after = TextMobject("After applying Linear Transformation")
         after.set_color(RED)
         after.move_to(3*UP+4.5*RIGHT)
-        after.scale(0.75)
+        after.scale(0.5)
         dot3 = Dot().shift(self.graph_origin+3*XTD*RIGHT+0*YTD*UP)
         dot4 = Dot().shift(self.graph_origin+4*XTD*RIGHT+1*YTD*UP)
         dot3.set_color(RED)
         dot4.set_color(RED)
-        p3 = TextMobject(r"$P_3$")
+        p3 = TextMobject(r"$T(P_1)$")
+        p3.scale(0.7)
         p3.set_color(RED)
         p3.move_to(self.graph_origin+3*XTD*RIGHT-1.1*YTD*UP)
-        p4 = TextMobject(r"$P_4$")
+        p4 = TextMobject(r"$T(P_2)$")
+        p4.scale(0.7)
         p4.set_color(RED)
-        p4.move_to(self.graph_origin+4*XTD*RIGHT+2*YTD*UP)
-
-        tp1 = TextMobject(r"$T(P_1) = P_3$")
-        tp1.set_color(RED)
-        tp1.move_to(DOWN+4*LEFT)
-        tp2 = TextMobject(r"$T(P_2) = P_4$")
-        tp2.set_color(RED)
-        tp2.move_to(2*DOWN+4*LEFT)
+        p4.move_to(self.graph_origin+4*XTD*RIGHT+1.5*YTD*UP)
 
         self.play(Write(before), ShowCreation(dot1), ShowCreation(dot2),Write(p1), Write(p2))
         self.wait(3)
-        self.play(Transform(before,after), Transform(dot1,dot3), Transform(dot2,dot4), Transform(p2,p4), Transform(p1,p3), Write(tp1), Write(tp2))
+        self.play(Transform(before,after), Transform(dot1,dot3), Transform(dot2,dot4), Transform(p2,p4), Transform(p1,p3))
         self.wait(3)
 
 
 class withgrid(LinearTransformationScene):
     def construct(self):
 
-        heading = TextMobject(r"$T(x,y) = (x+2y,x-y)$")
+        heading = TextMobject(r"Now, Imagine this happening for all the vectors")
+        heading.scale(0.5)
         heading.move_to(UP*2.5+LEFT*4)
         self.play(Write(heading))
         self.wait()
@@ -93,9 +92,11 @@ class withgrid(LinearTransformationScene):
         self.add_transformable_mobject(dot2_c)
         
         p1 = TextMobject(r"$P_1$")
+        p1.scale(0.75)
         p1.set_color(DARK_BLUE)
         p1.move_to(1*RIGHT+1.5*UP)
         p2 = TextMobject(r"$P_2$")
+        p2.scale(0.75)
         p2.set_color(DARK_BLUE)
         p2.move_to(2*RIGHT+1.5*UP)
 
@@ -107,10 +108,12 @@ class withgrid(LinearTransformationScene):
         dot4 = Dot().shift(4*RIGHT+1*UP)
         dot3.set_color(RED)
         dot4.set_color(RED)
-        p3 = TextMobject(r"$P_3$")
+        p3 = TextMobject(r"$T(P_1)$")
+        p3.scale(0.75)
         p3.set_color(RED)
         p3.move_to(3*RIGHT-0.6*UP)
-        p4 = TextMobject(r"$P_4$")
+        p4 = TextMobject(r"$T(P_2)$")
+        p4.scale(0.75)
         p4.set_color(RED)
         p4.move_to(4*RIGHT+1.5*UP)
 
@@ -118,6 +121,7 @@ class withgrid(LinearTransformationScene):
         self.wait(3)
         matrix = [[1,2],[1,-1]]
         self.apply_matrix(matrix)
+        self.play(FadeOut(dot1),FadeOut(dot2))
         self.play(Transform(before,after), Transform(p2,p4), Transform(p1,p3))
         self.play(Transform(before,after))
         self.wait(3)
@@ -141,17 +145,28 @@ class ThreeDExplanation(ThreeDScene):
 
         before = TextMobject("Before Linear Transformation")
         self.add_fixed_in_frame_mobjects(before)
-        before.set_color(GREEN_E)
+        before.set_color(YELLOW)
         before.move_to(3.5*UP+4*RIGHT)
         before.scale(0.75)
+
+        p1 = TextMobject(r"$P_1$")
+        p2 = TextMobject(r"$P_2$")
+        p3 = TextMobject(r"$P_3$")
+        p1.scale(0.75)
+        p2.scale(0.75)
+        p3.scale(0.75)
         dot1 = Dot().shift(1*RIGHT+1*UP)
         dot2 = Dot().shift(2*RIGHT+1*UP)
-        dot3 = Dot().shift(1*RIGHT+1*UP)
-        dot1.set_color(GREEN_E)
-        dot2.set_color(GREEN_E)
-        dot3.set_color(GREEN_E)
+        dot3 = Dot().shift(1*RIGHT+1*DOWN)
+        dot1.set_color(YELLOW)
+        dot2.set_color(YELLOW)
+        dot3.set_color(YELLOW)
         self.play(ShowCreation(before))
-
+        
+        p1.move_to(1*RIGHT+1*UP+[0,0,0.5])
+        p2.move_to(2*RIGHT+1*UP+[0,0,0.5])
+        p3.move_to(1*RIGHT-1*UP+[0,0,0.5])
+        
         dot1_c = Dot(radius = 0.05).shift(1*RIGHT+1*UP)
         dot2_c = Dot(radius = 0.05).shift(0*RIGHT+2*UP)
         dot3_c = Dot(radius = 0.05).shift(1*RIGHT-1*UP)
@@ -168,6 +183,9 @@ class ThreeDExplanation(ThreeDScene):
         self.stop_ambient_camera_rotation()
 
         plane = NumberPlane()
+        self.add_fixed_orientation_mobjects(p1)
+        self.add_fixed_orientation_mobjects(p2)
+        self.add_fixed_orientation_mobjects(p3)
         self.play(ShowCreation(dot1),ShowCreation(dot3),ShowCreation(dot2),ShowCreation(plane))
         
         self.play(FadeOut(before))
@@ -177,8 +195,23 @@ class ThreeDExplanation(ThreeDScene):
         after.move_to(3.5*UP+3.5*RIGHT)
         after.scale(0.75)
       
+        self.play(FadeOut(p1),FadeOut(p2),FadeOut(p3))
         matrix = [[1,1],[1,-1],[2,1]]
         self.play(FadeOut(dot1),FadeOut(dot2),FadeOut(dot3),ApplyMethod(plane.apply_matrix,matrix),ApplyMethod(dot1_c.apply_matrix,matrix),ApplyMethod(dot3_c.apply_matrix,matrix),ApplyMethod(dot2_c.apply_matrix,matrix))
+        
+        p4 = TextMobject(r"$T(P_1)$")
+        p5 = TextMobject(r"$T(P_2)$")
+        p6 = TextMobject(r"$T(P_3)$")
+        p4.scale(0.75)
+        p5.scale(0.75)
+        p6.scale(0.75)
+        p4.move_to(2*RIGHT+0*UP+[0,0,3.5])
+        p5.move_to(2*RIGHT-2*UP+[0,0,2.5])
+        p6.move_to(0*RIGHT+2*UP+[0,0,1.5])
+        self.add_fixed_orientation_mobjects(p5)
+        self.add_fixed_orientation_mobjects(p4)
+        self.add_fixed_orientation_mobjects(p6)
+        
         self.begin_ambient_camera_rotation(rate=0.3)
         self.wait(3)
         self.stop_ambient_camera_rotation()
@@ -188,4 +221,6 @@ class ThreeDExplanation(ThreeDScene):
         ending.move_to(-UP*2+LEFT*4)
         self.play(Transform(text,ending))
         self.add_fixed_in_frame_mobjects(ending)
+
+        self.play(FadeOut(plane))
         self.wait(9)