25 files changed, 463 insertions, 33 deletions
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/file1_introduction.py
index ccd23c9..ccd23c9 100644
--- 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/file1_introduction.py
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/file2_projections.py
index dd4b8d4..dd4b8d4 100755
--- 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/file2_projections.py
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/file3_orthonormal.py
index af51fc6..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/file3_orthonormal.py
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/file4_Non_Standard_Basis.py
index 6410a2c..6410a2c 100644
--- 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/file4_Non_Standard_Basis.py
diff --git a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file.txt b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file.txt
deleted file mode 100644
index cae98ce..0000000
--- a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file.txt
+++ /dev/null
@@ -1,3 +0,0 @@
-file 'text.mp4'
-file 'LinearTransformation.mp4'
-file 'NonLinearTransformation.mp4'
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)/file2_before_matrix.py
index 96e456d..96e456d 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)/file2_before_matrix.py
diff --git a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/square.py b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file3_square.py
index e828de4..e828de4 100644
--- a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/square.py
+++ b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file3_square.py
diff --git a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file2_Understand_Linear_Transformations_visually.py b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file4_Understand_Linear_Transformations_visually.py
index 577032d..577032d 100644
--- a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file2_Understand_Linear_Transformations_visually.py
+++ b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file4_Understand_Linear_Transformations_visually.py
diff --git a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file3_Uniform_Scaling.py b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file5_Uniform_Scaling.py
index a7856a5..a7856a5 100644
--- a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file3_Uniform_Scaling.py
+++ b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file5_Uniform_Scaling.py
diff --git a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file4_Horizontal_Shear.py b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file6_Horizontal_Shear.py
index 91f098e..91f098e 100644
--- a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file4_Horizontal_Shear.py
+++ b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file6_Horizontal_Shear.py
diff --git a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file4_Horizontal_Shear_gif.gif b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file6_Horizontal_Shear_gif.gif
index 9bef1b6..9bef1b6 100644
--- a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file4_Horizontal_Shear_gif.gif
+++ b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file6_Horizontal_Shear_gif.gif
diff --git a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file5_Vertical_Shear.py b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file7_Vertical_Shear.py
index 718e4e0..718e4e0 100644
--- a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file5_Vertical_Shear.py
+++ b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file7_Vertical_Shear.py
diff --git a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file5_Vertical_Shear_gif.gif b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file7_Vertical_Shear_gif.gif
index 7ca323f..7ca323f 100644
--- a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file5_Vertical_Shear_gif.gif
+++ b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file7_Vertical_Shear_gif.gif
diff --git a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file6_linear_transformation.py b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file8_linear_transformation.py
index 01a0cef..01a0cef 100755
--- a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file6_linear_transformation.py
+++ b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file8_linear_transformation.py
diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file1_Column_Space.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file10_NOT_in_lecture_note_Column_Space.py
index afe4f9a..afe4f9a 100644
--- a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file1_Column_Space.py
+++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file10_NOT_in_lecture_note_Column_Space.py
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/file1_Axb.py
index 95d1021..95d1021 100755
--- a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/Axb.py
+++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file1_Axb.py
diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file1_Column_Space.gif b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file1_Column_Space.gif
deleted file mode 100644
index 7d8d2e1..0000000
--- a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file1_Column_Space.gif
+++ /dev/null
diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/CSasImage.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file2_CSasImage.py
index fbb3291..70547cb 100644
--- a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/CSasImage.py
+++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file2_CSasImage.py
@@ -3,30 +3,31 @@ from manimlib.imports import *
 class Column_Space(Scene):
     def construct(self):
 
-        A = TextMobject(r"$A = $",r"$\left( \begin{array}{c c c} 1 & 2 & 1 \\ 1 & 3 & 1 \\ 2 & 1 & 4 \\ 3 & 2 & 3 \end{array} \right)$")
+        A = TextMobject(r"$A = $",r"$\left( \begin{array}{c c} 1 & 2 \\ 3 & 4 \end{array} \right)$")
         A.move_to(2*UP)
         A[1].set_color(color = DARK_BLUE)
         A.scale(0.75)
 
-        self.play(Write(A),run_time = 1)      
+        self.play(Write(A),run_time = 1)
 
-        CS_A = TextMobject(r"Column Space of $A = x_{1}$",r"$\left( \begin{array}{c} 1 \\ 1 \\ 2 \\ 3 \end{array} \right)$",r"$+x_{2}$",r"$ \left( \begin{array}{c} 2 \\ 3 \\ 1 \\ 2 \end{array} \right)$",r"$ + x_{3}$",r"$\left( \begin{array}{c} 1 \\ 1 \\ 4 \\ 3 \end{array} \right)$")
+        CS_A = TextMobject(r"Column Space of $A = x_{1}$",r"$\left( \begin{array}{c} 1 \\ 3 \end{array} \right)$",r"$+x_{2}$",r"$ \left( \begin{array}{c} 2 \\ 4\end{array} \right)$")
         CS_A.move_to(1.5*LEFT+1*DOWN)
         CS_A[1].set_color(color = DARK_BLUE)
         CS_A[3].set_color(color = DARK_BLUE)
-        CS_A[5].set_color(color = DARK_BLUE)
         CS_A.scale(0.75)
 
         self.play(Write(CS_A),run_time = 2)
 
-        arrow1 = Arrow(start = 1.25*UP,end = 0.25*DOWN+1.75*LEFT)
-        arrow2 = Arrow(start = 1.35*UP+0.5*RIGHT,end = 0.25*DOWN+0.5*RIGHT)
-        arrow3 = Arrow(start = 1.25*UP+0.75*RIGHT,end = 0.25*DOWN+2.9*RIGHT)
+        arrow1 = Arrow(start = 1.25*UP,end = (0.25*DOWN+1.75*LEFT+0.25*DOWN+1.2*RIGHT)/2)
+        arrow3 = Arrow(start = 1.25*UP+0.75*RIGHT,end = (0.25*DOWN+2.9*RIGHT+0.25*DOWN)/2)
+
+        arrow1.scale(1.5)
+        arrow3.scale(1.5)
 
         Defn = TextMobject("Linear Combination of Columns of Matrix")
         Defn.move_to(3*DOWN)
 
-        self.play(Write(Defn), ShowCreation(arrow1), ShowCreation(arrow2), ShowCreation(arrow3),run_time = 1)
+        self.play(Write(Defn), ShowCreation(arrow1), ShowCreation(arrow3),run_time = 1)
         self.wait(1)
 
 class solution(LinearTransformationScene):
@@ -43,7 +44,7 @@ class solution(LinearTransformationScene):
         self.wait()
         self.play(FadeOut(o))
 
-        A = TextMobject(r"Let $A$(= ",r"$\left[\begin{array}{c c} 1 & -1 \\ 1 & -1 \end{array}\right]$",r")denote the matrix the of this linear transformation.")
+        A = TextMobject(r"Let $A$ be ",r"$\left[\begin{array}{c c} 1 & -1 \\ 1 & -1 \end{array}\right]$",r". $A$ denotes the matrix the of this linear transformation.")
         A.move_to(2*DOWN)
         A.scale(0.75)
         A.add_background_rectangle()
@@ -116,8 +117,8 @@ class solution2nd(LinearTransformationScene):
         arrow2 = Arrow(start = ORIGIN,end = UP+LEFT)
         arrow3 = Arrow(start = ORIGIN,end = 3*UP+4*RIGHT)
         arrow1.set_color(YELLOW)
-        arrow2.set_color(YELLOW)
-        arrow3.set_color(YELLOW)
+        arrow2.set_color(ORANGE)
+        arrow3.set_color(PURPLE)
         arrow1.scale(1.3)
         arrow2.scale(1.5)
         arrow3.scale(1.1)
@@ -135,7 +136,7 @@ class solution2nd(LinearTransformationScene):
         self.wait()
         self.play(FadeOut(o))
 
-        A = TextMobject(r"Matrix the of this linear transformation is $A$(= ",r"$\left[\begin{array}{c c} 1 & -1 \\ 1 & -1 \end{array}\right]$",r") again.")
+        A = TextMobject(r"Let the matrix the of this linear transformation be $A$ =",r"$\left[\begin{array}{c c} 1 & -1 \\ 1 & -1 \end{array}\right]$",r" again.")
         A.move_to(2*DOWN)
         A.scale(0.75)
         A.add_background_rectangle()
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/file3_solution.py
index fb31881..eb310f3 100644
--- a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/solution.py
+++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file3_solution.py
@@ -5,16 +5,16 @@ class solution(LinearTransformationScene):
         self.setup()
         self.wait()
         
-        o = TextMobject(r"This is the original $2D$ vector space(before Linear Transformation)")
-        o.move_to(DOWN)
+        o = TextMobject(r"This is the original vector space $R^2$ (before Linear Transformation)")
+        o.move_to(3*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 = TextMobject(r"Consider the matrix the of this linear transformation $A$ = $\left[\begin{array}{c c} 1 & -1 \\  1 & -1 \end{array}\right]$")
+        A.move_to(3*DOWN)
         A.scale(0.75)
         A.add_background_rectangle()
         self.play(Write(A))
@@ -23,8 +23,8 @@ class solution(LinearTransformationScene):
         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 = TextMobject(r"This is the transformed vector space")
+        o.move_to(3*DOWN)
         o.scale(0.75)
         o.add_background_rectangle()
         self.play(Write(o))
@@ -32,15 +32,16 @@ class solution(LinearTransformationScene):
         self.play(FadeOut(o))
         
         arrow2 = Arrow(start = ORIGIN, end = 2*DOWN+2*LEFT)
-        arrow2.set_color(DARK_BLUE)
+        arrow2.set_color(PURPLE)
         arrow2.scale(1.2)
         self.play(ShowCreation(arrow2))
         self.wait()
 
-        o1 = TextMobject("If the vector lies in the transformed vector space")
+        o1 = TextMobject("If the ","vector b"," 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.move_to(2.5*DOWN+2*RIGHT)
+        o1[1].set_color(PURPLE)
+        o2.move_to(3*DOWN+2.5*RIGHT)
         o1.scale(0.75)
         o2.scale(0.75)
         o1.add_background_rectangle()
@@ -53,15 +54,16 @@ class solution(LinearTransformationScene):
         self.play(FadeOut(arrow2))
 
         arrow1 = Arrow(start = ORIGIN, end = 2*UP+RIGHT)
-        arrow1.set_color(DARK_BLUE)
+        arrow1.set_color(ORANGE)
         arrow1.scale(1.3)
         self.play(ShowCreation(arrow1))
         self.wait()
 
-        o1 = TextMobject("If the vector does lies in the transformed")
+        o1 = TextMobject("If the ","vector b"," 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.move_to(2.5*DOWN+2*RIGHT)
+        o1[1].set_color(ORANGE)
+        o2.move_to(3*DOWN+2.5*RIGHT)
         o1.scale(0.75)
         o2.scale(0.75)
         o1.add_background_rectangle()
diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/null_space.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file4_null_space.py
index dfc3cb4..3c52677 100644
--- a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/null_space.py
+++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file4_null_space.py
@@ -5,8 +5,8 @@ class null_space(LinearTransformationScene):
         self.setup()
         self.wait()
         
-        o = TextMobject(r"This is the original $2D$ vector space(before Linear Transformation)")
-        o.move_to(DOWN)
+        o = TextMobject(r"This is the original vector space $R^2$(before Linear Transformation)")
+        o.move_to(3*DOWN)
         o.scale(0.75)
         o.add_background_rectangle()
         self.play(Write(o))
@@ -64,7 +64,7 @@ class null_space(LinearTransformationScene):
         self.add_transformable_mobject(arrow6)
         self.add_transformable_mobject(arrow7)
 
-        o1 = TextMobject("Notice, entire set of vectors which belong to the vector")
+        o1 = TextMobject("Notice, entire set of vectors which belongs to the vector")
         o2 = TextMobject(r"subspace(Linear Span of $\left(\begin{array}{c} 1 \\ 1 \end{array}\right)$) transforms to zero")
         o1.move_to(2*DOWN+2.5*RIGHT)
         o2.move_to(2.75*DOWN+2.5*RIGHT)
@@ -83,7 +83,7 @@ class null_space(LinearTransformationScene):
         self.play(FadeOut(o1), FadeOut(o2))
 
         o = TextMobject(r"Hence, the vector space formed by linear span of $\left(\begin{array}{c} 1 \\ 1 \end{array}\right)$ is the null space of $A$")
-        o.move_to(DOWN)
+        o.move_to(3*DOWN)
         o.scale(0.75)
         o.add_background_rectangle()
         self.play(Write(o))
diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file5_Row_Space_part_1.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file5_Row_Space_part_1.py
new file mode 100644
index 0000000..5259eb4
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file5_Row_Space_part_1.py
@@ -0,0 +1,68 @@
+from manimlib.imports import *
+class LS(Scene):
+    def construct(self):
+        text1 = TextMobject(r"Consider a matrix $A =$")
+        text2 = TextMobject(r"[")
+        text3 = TextMobject(r"$\begin{array}{c c} 1 & -2\end{array}$")
+        text4 = TextMobject(r"$\begin{array}{c c} 1 & -1\end{array}$")
+        text5 = TextMobject(r"]")
+
+        text2.scale(2)
+        text5.scale(2)
+
+        text1.set_color(DARK_BLUE)
+        text2.set_color(DARK_BLUE)
+        text3.set_color(PURPLE)
+        text4.set_color(YELLOW)
+        text5.set_color(DARK_BLUE)
+
+        text1.move_to(3.5*LEFT+3*UP+2*RIGHT)
+        text2.move_to(0.75*LEFT+3*UP+2*RIGHT)
+        text3.move_to(3.25*UP+2*RIGHT)
+        text4.move_to(2.75*UP+2*RIGHT)
+        text5.move_to(0.75*RIGHT+3*UP+2*RIGHT)
+
+        self.play(FadeIn(text1), FadeIn(text2), FadeIn(text3), FadeIn(text4), FadeIn(text5))
+        self.wait()
+
+        ttext1 = TextMobject(r"$A^T =$")
+        ttext2 = TextMobject(r"[")
+        ttext3 = TextMobject(r"$\begin{array}{c} 1 \\ -2\end{array}$")
+        ttext4 = TextMobject(r"$\begin{array}{c} 1 \\ -1\end{array}$")
+        ttext5 = TextMobject(r"]")
+
+        ttext2.scale(2)
+        ttext5.scale(2)
+
+        ttext1.set_color(DARK_BLUE)
+        ttext2.set_color(DARK_BLUE)
+        ttext3.set_color(PURPLE)
+        ttext4.set_color(YELLOW)
+        ttext5.set_color(DARK_BLUE)
+
+        ttext1.move_to(2*LEFT+1.5*UP+2*RIGHT)
+        ttext2.move_to(1*LEFT+1.5*UP+2*RIGHT)
+        ttext3.move_to(0.5*LEFT+1.5*UP+2*RIGHT)
+        ttext4.move_to(0.5*RIGHT+1.5*UP+2*RIGHT)
+        ttext5.move_to(1*RIGHT+1.5*UP+2*RIGHT)
+
+        self.play(FadeIn(ttext1), FadeIn(ttext2), FadeIn(ttext3), FadeIn(ttext4), FadeIn(ttext5))
+
+        rtext = TextMobject(r"Row Space of $A$ = Column Space of $A^T = a_1$",r"$\left[\begin{array}{c} 1 \\ -2\end{array}\right]$",r"$+a_2$",r"$\left[\begin{array}{c} 1 \\ -1\end{array}\right]$")
+        rtext[1].set_color(PURPLE)
+        rtext[3].set_color(YELLOW)
+        rtext.move_to(2*DOWN+1.5*LEFT)
+        rtext.scale(0.75)
+
+        self.play(Write(rtext))
+        self.wait()
+
+        arrow1 = Arrow(start = 1.5*RIGHT+UP, end = 1.25*(DOWN+RIGHT))
+        arrow2 = Arrow(start = 2.5*RIGHT+UP, end = 1.25*DOWN+3.25*RIGHT)
+        arrow1.scale(1.25)
+        arrow2.scale(1.25)
+        arrow1.set_color(PURPLE)
+        arrow2.set_color(YELLOW)
+
+        self.play(ShowCreation(arrow1), ShowCreation(arrow2))
+        self.wait(2)
diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file2_Row_Space.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file6_Row_Space_part_2.py
index b16a32a..b16a32a 100644
--- a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file2_Row_Space.py
+++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file6_Row_Space_part_2.py
diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file7_Row_space_Orthogonal_Complements.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file7_Row_space_Orthogonal_Complements.py
new file mode 100644
index 0000000..c81d370
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file7_Row_space_Orthogonal_Complements.py
@@ -0,0 +1,150 @@
+from manimlib.imports import *
+class row_space(LinearTransformationScene):
+    def construct(self):
+
+        self.setup()
+        self.wait()
+        
+        o = TextMobject(r"This is the original vector space $R^2$(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))
+
+        o1 = TextMobject("Consider a set of vectors which are linear")
+        o2 = TextMobject(r"span of a $\left(\begin{array}{c} 1 \\ 1 \end{array}\right)$i.e. the null space.")
+        o1.move_to(2*DOWN+3*RIGHT)
+        o2.move_to(2.75*DOWN+3*RIGHT)
+        o1.scale(0.7)
+        o2.scale(0.7)
+        o1.add_background_rectangle()
+        o2.add_background_rectangle()
+        self.play(Write(o1))
+        self.play(Write(o2))
+
+        arrow = Arrow(start = ORIGIN, end = UP+RIGHT)
+        arrow.set_color(YELLOW)
+        arrow1 = Arrow(start = ORIGIN, end = 2*(UP+RIGHT))
+        arrow1.set_color(YELLOW)
+        arrow2 = Arrow(start = ORIGIN, end = 3*(UP+RIGHT))
+        arrow2.set_color(YELLOW)
+        arrow3 = Arrow(start = ORIGIN, end = 4*(UP+RIGHT))
+        arrow3.set_color(YELLOW)
+        arrow4 = Arrow(start = ORIGIN, end = DOWN+LEFT)
+        arrow4.set_color(YELLOW)
+        arrow5 = Arrow(start = ORIGIN, end = 2*(DOWN+LEFT))
+        arrow5.set_color(YELLOW)
+        arrow6 = Arrow(start = ORIGIN, end = 3*(DOWN+LEFT))
+        arrow6.set_color(YELLOW)
+        arrow7 = Arrow(start = ORIGIN, end = 4*(DOWN+LEFT))
+        arrow7.set_color(YELLOW)
+
+        arrow.scale(1.5)
+        arrow1.scale(1.2)
+        arrow2.scale(1.15)
+        arrow3.scale(1.1)
+        arrow4.scale(1.5)
+        arrow5.scale(1.2)
+        arrow6.scale(1.15)
+        arrow7.scale(1.1)
+
+        self.play(ShowCreation(arrow),
+        ShowCreation(arrow1),
+        ShowCreation(arrow2),
+        ShowCreation(arrow3),
+        ShowCreation(arrow4),
+        ShowCreation(arrow5),
+        ShowCreation(arrow6),
+        ShowCreation(arrow7),
+        )
+
+        self.wait(2)
+        self.play(FadeOut(o1), FadeOut(o2))
+
+        o1 = TextMobject("Consider a set of vectors which are linear")
+        o2 = TextMobject(r"span of a $\left(\begin{array}{c} 1 \\ -1 \end{array}\right)$i.e. the row space.")
+        o1.move_to(2*DOWN+3*RIGHT)
+        o2.move_to(2.75*DOWN+3*RIGHT)
+        o1.scale(0.7)
+        o2.scale(0.7)
+        o1.add_background_rectangle()
+        o2.add_background_rectangle()
+        self.play(Write(o1))
+        self.play(Write(o2))
+
+        rarrow = Arrow(start = ORIGIN, end = -UP+RIGHT)
+        rarrow.set_color(PURPLE)
+        rarrow1 = Arrow(start = ORIGIN, end = 2*(-UP+RIGHT))
+        rarrow1.set_color(PURPLE)
+        rarrow2 = Arrow(start = ORIGIN, end = 3*(-UP+RIGHT))
+        rarrow2.set_color(PURPLE)
+        rarrow3 = Arrow(start = ORIGIN, end = 4*(-UP+RIGHT))
+        rarrow3.set_color(PURPLE)
+        rarrow4 = Arrow(start = ORIGIN, end = -DOWN+LEFT)
+        rarrow4.set_color(PURPLE)
+        rarrow5 = Arrow(start = ORIGIN, end = 2*(-DOWN+LEFT))
+        rarrow5.set_color(PURPLE)
+        rarrow6 = Arrow(start = ORIGIN, end = 3*(-DOWN+LEFT))
+        rarrow6.set_color(PURPLE)
+        rarrow7 = Arrow(start = ORIGIN, end = 4*(-DOWN+LEFT))
+        rarrow7.set_color(PURPLE)
+
+        rarrow.scale(1.5)
+        rarrow1.scale(1.2)
+        rarrow2.scale(1.15)
+        rarrow3.scale(1.1)
+        rarrow4.scale(1.5)
+        rarrow5.scale(1.2)
+        rarrow6.scale(1.15)
+        rarrow7.scale(1.1)
+
+        self.play(ShowCreation(rarrow),
+        ShowCreation(rarrow1),
+        ShowCreation(rarrow2),
+        ShowCreation(rarrow3),
+        ShowCreation(rarrow4),
+        ShowCreation(rarrow5),
+        ShowCreation(rarrow6),
+        ShowCreation(rarrow7),
+        )
+
+        self.wait(2)
+        self.play(FadeOut(o1), FadeOut(o2))
+
+        self.add_transformable_mobject(arrow)
+        self.add_transformable_mobject(arrow1)
+        self.add_transformable_mobject(arrow2)
+        self.add_transformable_mobject(arrow3)
+        self.add_transformable_mobject(arrow4)
+        self.add_transformable_mobject(arrow5)
+        self.add_transformable_mobject(arrow6)
+        self.add_transformable_mobject(arrow7)
+
+        self.add_transformable_mobject(rarrow)
+        self.add_transformable_mobject(rarrow1)
+        self.add_transformable_mobject(rarrow2)
+        self.add_transformable_mobject(rarrow3)
+        self.add_transformable_mobject(rarrow4)
+        self.add_transformable_mobject(rarrow5)
+        self.add_transformable_mobject(rarrow6)
+        self.add_transformable_mobject(rarrow7)
+
+        o1 = TextMobject("Notice, entire set of vectors which belong to the null space of $A$ transforms to zero")
+        o2 = TextMobject(r"and entire set of vectors which belong to the row space of $A$ transforms to column space of $A$.")
+        o1.move_to(2.5*DOWN)
+        o2.move_to(3.5*DOWN)
+        o1.scale(0.7)
+        o2.scale(0.7)
+        o1.add_background_rectangle()
+        o2.add_background_rectangle()
+        self.play(Write(o1))
+        self.play(Write(o2))
+        self.wait()
+
+        matrix = [[1,-1],[1,-1]]
+        self.apply_matrix(matrix)
+        self.wait(3)
+
+        self.play(FadeOut(o1), FadeOut(o2))
+\ No newline at end of file
diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file8_Left_Null_Space.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file8_Left_Null_Space.py
new file mode 100755
index 0000000..fd05e75
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file8_Left_Null_Space.py
@@ -0,0 +1,26 @@
+from manimlib.imports import *
+
+class Left_Null_Space(Scene):
+    def construct(self):
+
+        A = TextMobject(r"Left Null Space of A")
+        A.move_to(3*UP)
+        defn = TextMobject(r"It is a vector space that consists of all the solution $x$ to the equation $A^{T}x=0$")
+        defn.move_to(2*UP)
+        defn.scale(0.75)
+        eqn1 = TextMobject(r"$A^{T}x=0 \cdots (i)$")
+        eqn1.move_to(UP)
+        self.play(Write(A), Write(defn), Write(eqn1),run_time=1)
+        statement = TextMobject(r"Taking transpose of eqn $(i)$")
+        eqn = TextMobject(r"$(A^{T}x)^{T}=0$")
+        eqn.move_to(DOWN)
+        eqn2 = TextMobject(r"$x^{T}(A^{T})^{T}=0$")
+        eqn2.move_to(DOWN)
+        eqn3 = TextMobject(r"$x^{T}A=0$")
+        eqn3.move_to(DOWN)
+        self.play(Write(statement),Write(eqn),run_time=1)
+        self.wait(0.5)
+        self.play(Transform(eqn,eqn2),run_time=1)
+        self.wait(0.5)
+        self.play(Transform(eqn,eqn3),run_time=1)
+        self.wait(0.5)
+\ No newline at end of file
diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file9_left_null_space.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file9_left_null_space.py
new file mode 100644
index 0000000..61285be
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/file9_left_null_space.py
@@ -0,0 +1,186 @@
+from manimlib.imports import *
+class row_space(LinearTransformationScene):
+    def construct(self):
+
+        self.setup()
+        self.wait()
+        
+        o = TextMobject(r"This is the original vector space $R^2$(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))
+
+        o1 = TextMobject("Consider a set of vectors which are linear")
+        o2 = TextMobject(r"span of a $\left(\begin{array}{c} 1 \\ 1 \end{array}\right)$i.e. the null space.")
+        o1.move_to(2*DOWN+3*RIGHT)
+        o2.move_to(2.75*DOWN+3*RIGHT)
+        o1.scale(0.7)
+        o2.scale(0.7)
+        o1.add_background_rectangle()
+        o2.add_background_rectangle()
+        self.play(Write(o1))
+        self.play(Write(o2))
+
+        arrow = Arrow(start = ORIGIN, end = UP+RIGHT)
+        arrow.set_color(YELLOW)
+        arrow1 = Arrow(start = ORIGIN, end = 2*(UP+RIGHT))
+        arrow1.set_color(YELLOW)
+        arrow2 = Arrow(start = ORIGIN, end = 3*(UP+RIGHT))
+        arrow2.set_color(YELLOW)
+        arrow3 = Arrow(start = ORIGIN, end = 4*(UP+RIGHT))
+        arrow3.set_color(YELLOW)
+        arrow4 = Arrow(start = ORIGIN, end = DOWN+LEFT)
+        arrow4.set_color(YELLOW)
+        arrow5 = Arrow(start = ORIGIN, end = 2*(DOWN+LEFT))
+        arrow5.set_color(YELLOW)
+        arrow6 = Arrow(start = ORIGIN, end = 3*(DOWN+LEFT))
+        arrow6.set_color(YELLOW)
+        arrow7 = Arrow(start = ORIGIN, end = 4*(DOWN+LEFT))
+        arrow7.set_color(YELLOW)
+
+        arrow.scale(1.5)
+        arrow1.scale(1.2)
+        arrow2.scale(1.15)
+        arrow3.scale(1.1)
+        arrow4.scale(1.5)
+        arrow5.scale(1.2)
+        arrow6.scale(1.15)
+        arrow7.scale(1.1)
+
+        self.play(ShowCreation(arrow),
+        ShowCreation(arrow1),
+        ShowCreation(arrow2),
+        ShowCreation(arrow3),
+        ShowCreation(arrow4),
+        ShowCreation(arrow5),
+        ShowCreation(arrow6),
+        ShowCreation(arrow7),
+        )
+
+        self.wait(2)
+        self.play(FadeOut(o1), FadeOut(o2))
+
+        o1 = TextMobject("Consider a set of vectors which are linear")
+        o2 = TextMobject(r"span of a vector $\left(\begin{array}{c} 1 \\ -1 \end{array}\right)$i.e. the row space.")
+        o1.move_to(2*DOWN+3*RIGHT)
+        o2.move_to(2.75*DOWN+3*RIGHT)
+        o1.scale(0.7)
+        o2.scale(0.7)
+        o1.add_background_rectangle()
+        o2.add_background_rectangle()
+        self.play(Write(o1))
+        self.play(Write(o2))
+
+        rarrow = Arrow(start = ORIGIN, end = -UP+RIGHT)
+        rarrow.set_color(PURPLE)
+        rarrow1 = Arrow(start = ORIGIN, end = 2*(-UP+RIGHT))
+        rarrow1.set_color(PURPLE)
+        rarrow2 = Arrow(start = ORIGIN, end = 3*(-UP+RIGHT))
+        rarrow2.set_color(PURPLE)
+        rarrow3 = Arrow(start = ORIGIN, end = 4*(-UP+RIGHT))
+        rarrow3.set_color(PURPLE)
+        rarrow4 = Arrow(start = ORIGIN, end = -DOWN+LEFT)
+        rarrow4.set_color(PURPLE)
+        rarrow5 = Arrow(start = ORIGIN, end = 2*(-DOWN+LEFT))
+        rarrow5.set_color(PURPLE)
+        rarrow6 = Arrow(start = ORIGIN, end = 3*(-DOWN+LEFT))
+        rarrow6.set_color(PURPLE)
+        rarrow7 = Arrow(start = ORIGIN, end = 4*(-DOWN+LEFT))
+        rarrow7.set_color(PURPLE)
+
+        rarrow.scale(1.5)
+        rarrow1.scale(1.2)
+        rarrow2.scale(1.15)
+        rarrow3.scale(1.1)
+        rarrow4.scale(1.5)
+        rarrow5.scale(1.2)
+        rarrow6.scale(1.15)
+        rarrow7.scale(1.1)
+
+        self.play(ShowCreation(rarrow),
+        ShowCreation(rarrow1),
+        ShowCreation(rarrow2),
+        ShowCreation(rarrow3),
+        ShowCreation(rarrow4),
+        ShowCreation(rarrow5),
+        ShowCreation(rarrow6),
+        ShowCreation(rarrow7),
+        )
+
+        self.wait(2)
+        self.play(FadeOut(o1), FadeOut(o2))
+
+        self.add_transformable_mobject(arrow)
+        self.add_transformable_mobject(arrow1)
+        self.add_transformable_mobject(arrow2)
+        self.add_transformable_mobject(arrow3)
+        self.add_transformable_mobject(arrow4)
+        self.add_transformable_mobject(arrow5)
+        self.add_transformable_mobject(arrow6)
+        self.add_transformable_mobject(arrow7)
+
+        self.add_transformable_mobject(rarrow)
+        self.add_transformable_mobject(rarrow1)
+        self.add_transformable_mobject(rarrow2)
+        self.add_transformable_mobject(rarrow3)
+        self.add_transformable_mobject(rarrow4)
+        self.add_transformable_mobject(rarrow5)
+        self.add_transformable_mobject(rarrow6)
+        self.add_transformable_mobject(rarrow7)
+
+        matrix = [[1,-1],[1,-1]]
+        self.apply_matrix(matrix)
+        self.wait(3)
+
+        o1 = TextMobject("Consider a set of vectors which are linear span of a vector")
+        o2 = TextMobject(r"$\left(\begin{array}{c} 1 \\ -1 \end{array}\right)$ which is orthogonal to column space i.e. Left Null Space")
+        o1.move_to(2*DOWN)
+        o2.move_to(2.75*DOWN)
+        o1.scale(0.7)
+        o2.scale(0.7)
+        o1.add_background_rectangle()
+        o2.add_background_rectangle()
+        self.play(Write(o1))
+        self.play(Write(o2))
+
+        rarrow = Arrow(start = ORIGIN, end = -UP+RIGHT)
+        rarrow.set_color(YELLOW)
+        rarrow1 = Arrow(start = ORIGIN, end = 2*(-UP+RIGHT))
+        rarrow1.set_color(YELLOW)
+        rarrow2 = Arrow(start = ORIGIN, end = 3*(-UP+RIGHT))
+        rarrow2.set_color(YELLOW)
+        rarrow3 = Arrow(start = ORIGIN, end = 4*(-UP+RIGHT))
+        rarrow3.set_color(YELLOW)
+        rarrow4 = Arrow(start = ORIGIN, end = -DOWN+LEFT)
+        rarrow4.set_color(YELLOW)
+        rarrow5 = Arrow(start = ORIGIN, end = 2*(-DOWN+LEFT))
+        rarrow5.set_color(YELLOW)
+        rarrow6 = Arrow(start = ORIGIN, end = 3*(-DOWN+LEFT))
+        rarrow6.set_color(YELLOW)
+        rarrow7 = Arrow(start = ORIGIN, end = 4*(-DOWN+LEFT))
+        rarrow7.set_color(YELLOW)
+
+        rarrow.scale(1.5)
+        rarrow1.scale(1.2)
+        rarrow2.scale(1.15)
+        rarrow3.scale(1.1)
+        rarrow4.scale(1.5)
+        rarrow5.scale(1.2)
+        rarrow6.scale(1.15)
+        rarrow7.scale(1.1)
+
+        self.play(ShowCreation(rarrow),
+        ShowCreation(rarrow1),
+        ShowCreation(rarrow2),
+        ShowCreation(rarrow3),
+        ShowCreation(rarrow4),
+        ShowCreation(rarrow5),
+        ShowCreation(rarrow6),
+        ShowCreation(rarrow7),
+        )
+
+        self.wait(2)
+        self.play(FadeOut(o1), FadeOut(o2))