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diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Basis_of_a_dual_vector_space.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Basis_of_a_dual_vector_space.py
new file mode 100644
index 0000000..630670e
--- /dev/null
+++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Basis_of_a_dual_vector_space.py
@@ -0,0 +1,126 @@
+from manimlib.imports import *

+class DualVectorSpace(ZoomedScene):

+	

+	def construct(self):

+		c1 = Ellipse(radius = 2,color=BLUE)

+		c2 = Ellipse(radius = 2,color=YELLOW)

+		c1.rotate(np.pi/2)

+		c2.rotate(np.pi/2)

+		c1.shift(2*LEFT+0.6*UP)

+		c2.shift(2*RIGHT+0.6*UP)

+		c1.scale(2)

+		c2.scale(2)

+		self.play(ShowCreation(c1))

+		self.play(ShowCreation(c2))

+		dot1 = SmallDot(color=BLUE).shift(2*LEFT+2*UP)

+		dot2 = SmallDot(color=BLUE).shift(2*LEFT+1.5*UP)

+		dot3 = SmallDot(color=BLUE).shift(2*LEFT+1*UP)

+		dot4 = SmallDot(color=BLUE).shift(2*LEFT+0.5*UP)

+		dot5 = SmallDot(color=BLUE).shift(2*LEFT)

+		dot6 = SmallDot(color=BLUE).shift(2*LEFT+0.5*DOWN)

+		dot7 = SmallDot(color=BLUE).shift(2*LEFT+1*DOWN)

+		text1 = TextMobject(r"$V$").scale(0.6).shift(3*UP+2*LEFT)

+		text2 = TextMobject(r"$V^* = \{T:V\rightarrow F\}$").scale(0.6).shift(3*UP+2.5*RIGHT)		

+		self.play(ShowCreation(dot1),ShowCreation(dot2),ShowCreation(dot3),ShowCreation(dot4),ShowCreation(dot5),ShowCreation(dot6),ShowCreation(dot7))

+		v1 = TextMobject(r"$v_1$").scale(0.5).shift(2.2*LEFT+2*UP)

+		v2 = TextMobject(r"$v_2$").scale(0.5).shift(2.2*LEFT+1.5*UP)

+		v3 = TextMobject(r"$v_3$").scale(0.5).shift(2.2*LEFT+1*UP)

+		v4 = TextMobject(r"$v_4$").scale(0.5).shift(2.2*LEFT+0.5*UP)

+		v5 = TextMobject(r"$v_5$").scale(0.5).shift(2.2*LEFT)

+		v6 = TextMobject(r"$v_6$").scale(0.5).shift(2.2*LEFT+0.5*DOWN)

+		v7 = TextMobject(r"$v_7$").scale(0.5).shift(2.2*LEFT+1*DOWN)			

+		self.play(ShowCreation(v1),ShowCreation(v2),ShowCreation(v3),ShowCreation(v4),ShowCreation(v5),ShowCreation(v6),ShowCreation(v7))

+		self.play(ShowCreation(text1))

+		dot9 = SmallDot(color=YELLOW).shift(2*RIGHT+2*UP)

+		dot10 = SmallDot(color=YELLOW).shift(2*RIGHT+1.5*UP)

+		dot11 = SmallDot(color=YELLOW).shift(2*RIGHT+1*UP)

+		dot12 = SmallDot(color=YELLOW).shift(2*RIGHT+0.5*UP)

+		dot13 = SmallDot(color=YELLOW).shift(2*RIGHT)

+		dot14 = SmallDot(color=YELLOW).shift(2*RIGHT+0.5*DOWN)

+		dot15 = SmallDot(color=YELLOW).shift(2*RIGHT+1*DOWN)		

+		self.play(ShowCreation(dot9),ShowCreation(dot10),ShowCreation(dot11),ShowCreation(dot12),ShowCreation(dot13),ShowCreation(dot14),ShowCreation(dot15))

+		v9 = TextMobject(r"${T_1}$").scale(0.5).shift(2.2*RIGHT+2*UP)

+		v10 = TextMobject(r"${T_2}$").scale(0.5).shift(2.2*RIGHT+1.5*UP)

+		v11 = TextMobject(r"${T_3}$").scale(0.5).shift(2.2*RIGHT+1*UP)

+		v12 = TextMobject(r"${T_4}$").scale(0.5).shift(2.2*RIGHT+0.5*UP)

+		v13 = TextMobject(r"${T_5}$").scale(0.5).shift(2.2*RIGHT)

+		v14 = TextMobject(r"${T_6}$").scale(0.5).shift(2.2*RIGHT+0.5*DOWN)

+		v15 = TextMobject(r"${T_7}$").scale(0.5).shift(2.2*RIGHT+1*DOWN)		

+		self.play(ShowCreation(v9),ShowCreation(v10),ShowCreation(v11),ShowCreation(v12),ShowCreation(v13),ShowCreation(v14),ShowCreation(v15))

+		self.play(ShowCreation(text2))

+		line1 = Line(start=dot1,end=dot9,stroke_width=0.95)

+		line2 = Line(start=dot2,end=dot10,stroke_width=0.95)

+		line3 = Line(start=dot3,end=dot11,stroke_width=0.95)

+		line4 = Line(start=dot4,end=dot12,stroke_width=0.95)

+		line5 = Line(start=dot5,end=dot13,stroke_width=0.95)

+		line6 = Line(start=dot6,end=dot14,stroke_width=0.95)

+		line7 = Line(start=dot7,end=dot15,stroke_width=0.95)		

+		self.play(ShowCreation(line1),ShowCreation(line2),ShowCreation(line3),ShowCreation(line4),ShowCreation(line5),ShowCreation(line6),ShowCreation(line7))

+		self.wait(1.5)

+		rect1 = Rectangle(stroke_width=0.5,width=1,height=1.9).set_fill(color=BLUE,opacity=0.3)

+		vgroup1 = VGroup(dot3,dot4,v3,v4)

+		rect1.surround(vgroup1)

+		self.play(ShowCreation(rect1))

+		text3 = TextMobject(r"Basis of $V$ = $\{v_3, v_4\}$").shift(4.6*LEFT+1*UP).scale(0.5)

+		self.play(ShowCreation(text3))

+		self.wait(1.5)

+		rect2 = Rectangle(stroke_width=0.5,width=1,height=1.7).set_fill(color=YELLOW,opacity=0.3)

+		vgroup2 = VGroup(dot11,dot12,v11,v12)

+		rect2.surround(vgroup2)

+		self.play(ShowCreation(rect2))

+		text4 = TextMobject(r"Basis of $V^*$ = $\{{T_3},{T_4} \}$").shift(4.6*RIGHT+1*UP).scale(0.5)

+		self.play(ShowCreation(text4))

+		self.wait(2.5)

+		v9.move_to(3*LEFT+3*UP).scale(1.2).set_color(YELLOW)

+		colon = TextMobject(":").shift(3*UP+2.6*LEFT)

+		vgroup3 = VGroup(line1,line2,line3,line4,line5,line6,line7)

+		vgroup4 = VGroup(v10,v11,v12,v13,v14,v15,rect1,rect2,text3,text4,c2)

+		vgroup5 = VGroup(dot9,dot10,dot11,dot12,dot13,dot14,dot15)

+		text5 = TextMobject(r"$F$").scale(0.6).shift(3*UP+2*RIGHT)

+		dot9 = SmallDot(color=GREEN).shift(2*RIGHT+2*UP)

+		dot10 = SmallDot(color=GREEN).shift(2*RIGHT+1.5*UP)

+		dot11 = SmallDot(color=GREEN).shift(2*RIGHT+1*UP)

+		dot12 = SmallDot(color=GREEN).shift(2*RIGHT+0.5*UP)

+		dot13 = SmallDot(color=GREEN).shift(2*RIGHT)

+		dot14 = SmallDot(color=GREEN).shift(2*RIGHT+0.5*DOWN)

+		dot15 = SmallDot(color=GREEN).shift(2*RIGHT+1*DOWN)		

+		f1 = TextMobject(r"${f_1}$").scale(0.5).shift(2.2*RIGHT+2*UP)

+		f2 = TextMobject(r"${f_2}$").scale(0.5).shift(2.2*RIGHT+1.5*UP)

+		f3 = TextMobject(r"${f_3}$").scale(0.5).shift(2.2*RIGHT+1*UP)

+		f4 = TextMobject(r"${f_4}$").scale(0.5).shift(2.2*RIGHT+0.5*UP)

+		f5 = TextMobject(r"${f_5}$").scale(0.5).shift(2.2*RIGHT)

+		f6 = TextMobject(r"${f_6}$").scale(0.5).shift(2.2*RIGHT+0.5*DOWN)

+		f7 = TextMobject(r"${f_7}$").scale(0.5).shift(2.2*RIGHT+1*DOWN)

+		vgroup6 = VGroup(f1,f2,f3,f4,f5,f6,f7)

+		arrow = Arrow(stroke_width=1.6).scale(1.5).shift(3*UP)

+		c3 = Ellipse(radius = 2,color=GREEN)

+		c3.rotate(np.pi/2)		

+		c3.shift(2*RIGHT+0.6*UP)		

+		c3.scale(2)		

+		self.play(ShowCreation(v9))

+		self.wait(1.5)

+		self.play(ShowCreation(arrow),ShowCreation(colon),Transform(text2,text5),FadeOut(vgroup3),FadeOut(vgroup4),FadeOut(vgroup5))

+		self.play(ShowCreation(vgroup5),ShowCreation(vgroup6),ShowCreation(c3))

+		self.wait(0.7)

+		self.play(ShowCreation(vgroup3))

+		self.wait(3)

+

+		

+		

+

+		 

+		

+

+

+

+

+

+

+

+

+		

+	

+

+

+

+

diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Dual_Basis_Example.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Dual_Basis_Example.py
new file mode 100644
index 0000000..d79ec3e
--- /dev/null
+++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Dual_Basis_Example.py
@@ -0,0 +1,97 @@
+from manimlib.imports import *

+import numpy as np

+

+class Dual_Basis(GraphScene):

+  CONFIG={

+  "x_min": -7,

+  "x_max": 7,

+  "y_min": -7,

+  "y_max": 7,

+  "graph_origin": ORIGIN,

+  "x_axis_label":"$X$",

+  "y_axis_label":"$Y$",

+  "x_labeled_nums": list(np.arange(-7, 8,1)),

+  "y_labeled_nums": list(np.arange(-7, 8,1)),

+  "x_axis_width": 10,

+  "y_axis_height": 10,

+  "x_tick_frequency":1,

+  "axes_color": GREY,

+  "area_opacity": 3,

+  "num_rects": 10,

+  } 

+  def construct(self):

+    self.setup_axes(animate = True)

+    XD = self.x_axis_width/(self.x_max- self.x_min)

+    YD = self.y_axis_height/(self.y_max- self.y_min)  

+    a1=2*XD*RIGHT+1*YD*UP

+    a2=3*XD*RIGHT+1*YD*UP

+    vec1=Vector(direction=a1,stroke_width=2).set_color(RED_E)

+    vec1.shift(self.graph_origin)    

+    v1_label=TextMobject(r"$v_1$")

+    v1_label=(v1_label.shift(self.graph_origin+a1+0.1)).scale(.6)

+    self.play(ShowCreation(vec1),ShowCreation(v1_label))

+    text1=TextMobject(r"\text{$v_1$}",r"\text{$= (2,1)$}").scale(.6)

+    text1[0].set_color(RED_E)

+    text1.shift(5*LEFT+3.5*UP)

+    self.play(ShowCreation(text1))

+    self.wait(1.5)

+    vec2=Vector(direction=a2,stroke_width=2).set_color(YELLOW_E)

+    vec2.shift(self.graph_origin)

+    v2_label=TextMobject(r"$v_2$")

+    v2_label=(v2_label.shift(self.graph_origin+a2+0.1)).scale(.6)

+    self.play(ShowCreation(vec2),ShowCreation(v2_label))

+    text2=TextMobject(r"\text{$v_2$}",r"\text{$= (3,1)$}").scale(.6)

+    text2[0].set_color(YELLOW_E)

+    text2.shift(5*LEFT+3*UP)

+    self.play(ShowCreation(text2))

+    self.wait(1.5)

+    text3=TextMobject(r"\text{${T_2}$}",r"\text{$(v_1)$}",r"\text{$= 0$}").scale(.6)

+    text3[0].set_color(BLUE)

+    text3[1].set_color(RED_E)

+    text3.shift(4.94*LEFT+2.5*UP)

+    self.play(ShowCreation(text3))

+    self.wait(1.5)

+    text4=TextMobject(r"\text{${T_2}$}",r"\text{$= x - 2y$}").scale(.6)

+    text4[0].set_color(BLUE)

+    text4.shift(4.9*LEFT+2*UP)

+    self.play(ShowCreation(text4))

+    self.wait(1.5)

+    line1 = self.get_graph(lambda x : x/2, x_min = -5,x_max=5,color=BLUE)

+    v1_dual_label = TextMobject(r"${T_2}$").scale(.6).shift(3.9*RIGHT+1.85*UP)    

+    self.play(ShowCreation(line1),ShowCreation(v1_dual_label))

+    self.wait(1.5)    

+    text5=TextMobject(r"\text{${T_1}$}",r"\text{$(v_2)$}",r"\text{$= 0$}").scale(.6)

+    text5[1].set_color(YELLOW_E)

+    text5[0].set_color(PINK)

+    text5.shift(4.94*LEFT+1.5*UP)

+    self.play(ShowCreation(text5))

+    self.wait(1.5)

+    line2 = self.get_graph(lambda x : x/3, x_min = -5,x_max=5,color=PINK)

+    v2_dual_label = TextMobject(r"${T_1}$").scale(.6).shift(3.9*RIGHT+1.3*UP)

+    self.play(ShowCreation(line2),ShowCreation(v2_dual_label))

+    self.wait(1.5)

+    text6=TextMobject(r"\text{${T_1}$}",r"\text{$= - x + 3y$}").scale(.6)

+    text6[0].set_color(PINK)

+    text6.shift(4.76*LEFT+1*UP)

+    self.play(ShowCreation(text6))

+    self.wait(3)

+    text7 = TextMobject(r"\text{B =}",r"\text{$[$}",r"\text{$v_1,$}",r"\text{$v_2$}",r"\text{$]$}",r"\text{=}",r"\text{$[(2,1), (3,1)]$}").scale(0.6).shift(3*UP+4.5*LEFT)

+    text7[2].set_color(RED_E)

+    text7[3].set_color(YELLOW_E)

+    self.play(FadeOut(text1),FadeOut(text2),FadeOut(text3),FadeOut(text4),FadeOut(text5),FadeOut(text6))

+    self.play(ShowCreation(text7))

+    self.wait(0.7)

+    text8 = TextMobject(r"\text{B$^* =$}",r"\text{$[$}",r"\text{${T_1}$,}",r"\text{${T_2} $}",r"\text{$]$}",r"\text{=}",r"\text{$[-x + 3y, x - 2y]$}").scale(0.6).shift(2.3*UP+4.1*LEFT)

+    text8[3].set_color(BLUE)

+    text8[2].set_color(PINK)

+    self.play(ShowCreation(text8))

+    self.wait(3)

+

+

+

+

+ 

+

+

+   

+

diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Dual_of_a_Cube.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Dual_of_a_Cube.py
new file mode 100644
index 0000000..a6f501e
--- /dev/null
+++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Dual_of_a_Cube.py
@@ -0,0 +1,38 @@
+from manimlib.imports import *

+class Duality(ThreeDScene):

+

+

+    def construct(self):

+        axes = ThreeDAxes()  

+        self.set_camera_orientation(phi = 65*DEGREES,theta =80*DEGREES)

+        self.begin_ambient_camera_rotation(rate=0.09)     

+        cube = Cube(stroke_width=5,color=WHITE).scale(2)      

+        cube.set_opacity(0.2) 

+        self.play(ShowCreation(cube))        

+        dot1= Dot(color=RED).scale(0.85).shift([2,0,0])        

+        self.play(ShowCreation(dot1))

+        dot2= Dot(color=YELLOW).scale(0.85).shift([-2,0,0])        

+        self.play(ShowCreation(dot2))

+        dot3= Dot(color=BLUE).scale(0.85).shift([0,-2,0])        

+        self.play(ShowCreation(dot3))

+        dot4= Dot(color=GREEN).scale(0.85).shift([0,2,0])       

+        self.play(ShowCreation(dot4))

+        dot5= Dot(color=ORANGE).scale(0.85).shift([0,0,2])        

+        self.play(ShowCreation(dot5))

+        dot6= Dot(color=PINK).scale(0.85).shift([0,0,-2])       

+        self.play(ShowCreation(dot6))

+        line1 = Line(start=[0,0,2],end=[2,0,0],stroke_width=2.5,color=BLACK)        

+        line2 = Line(start=[0,0,2],end=[-2,0,0],stroke_width=2.5,color=BLACK)        

+        line3 = Line(start=[0,0,2],end=[0,-2,0],stroke_width=2.5,color=BLACK)        

+        line4 = Line(start=[0,0,2],end=[0,2,0],stroke_width=2.5,color=BLACK)        

+        line5 = Line(start=[2,0,0],end=[0,0,-2],stroke_width=2.5,color=BLACK)        

+        line6 = Line(start=[-2,0,0],end=[0,0,-2],stroke_width=2.5,color=BLACK)        

+        line7 = Line(start=[0,-2,0],end=[0,0,-2],stroke_width=2.5,color=BLACK)        

+        line8 = Line(start=[0,2,0],end=[0,0,-2],stroke_width=2.5,color=BLACK)        

+        line9 = Line(start=[0,2,0],end=[-2,0,0],stroke_width=2.5,color=BLACK)        

+        line10 = Line(start=[-2,0,0],end=[0,-2,0],stroke_width=2.5,color=BLACK)        

+        line11 = Line(start=[0,-2,0],end=[2,0,0],stroke_width=2.5,color=BLACK)        

+        line12 = Line(start=[2,0,0],end=[0,2,0],stroke_width=2.5,color=BLACK)        

+        self.play(ShowCreation(line1),ShowCreation(line2),ShowCreation(line3),ShowCreation(line4),ShowCreation(line5),ShowCreation(line6),ShowCreation(line7),ShowCreation(line8),ShowCreation(line9),ShowCreation(line10),ShowCreation(line11),ShowCreation(line12))

+        self.wait(10)

+       

diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Duality_in_Sets.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Duality_in_Sets.py
new file mode 100644
index 0000000..693017e
--- /dev/null
+++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Duality_in_Sets.py
@@ -0,0 +1,25 @@
+from manimlib.imports import *

+import numpy as np

+class Duality_in_sets(Scene):

+	def construct(self):

+		circle1 = Circle(radius=0.4,color=BLACK).shift(2.3*LEFT)

+		circle1.set_fill(color=RED,opacity=200)

+		rect1=Rectangle(height=2,width=2,color=GREY).shift(2*LEFT)

+		rect1.set_fill(color=DARK_BLUE,opacity=1)		

+		text1 = TextMobject("S").scale(0.7).shift(0.9*UP+0.7*LEFT)

+		text2 = TextMobject("X",color=BLACK,stroke_width=0.5).scale(0.5).shift(2.3*LEFT)

+		self.play(ShowCreation(rect1),ShowCreation(text1),ShowCreation(circle1),ShowCreation(text2))

+		circle2 = Circle(radius=0.4,color=BLACK).shift(1.7*RIGHT)

+		circle2.set_fill(color=BLACK,opacity=200)

+		rect2=Rectangle(height=2,width=2,color=GREY).shift(2*RIGHT)

+		rect2.set_fill(color=DARK_BLUE,opacity=1)	

+		text3 = TextMobject("S").scale(0.7).shift(0.9*UP+3.3*RIGHT)

+		text4 = TextMobject(r"X$^c$",color=BLACK,stroke_width=0.2).scale(0.5).shift(2.55*RIGHT+0.6*UP)

+		text5 = TextMobject(r"\text{The subset}",r"\text{X$^c$}",r"\text{is the dual of subset}",r"\text{X}").scale(0.6).shift(2.7*UP+0.5*LEFT)

+		text5[1].set_color(GREY)

+		text5[3].set_color(GREY)

+		self.play(ShowCreation(rect2),ShowCreation(circle2),ShowCreation(text3),ShowCreation(text4))

+		self.wait(2)

+		self.play(ShowCreation(text5))

+		self.wait(3)

+		

diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Linear_Functional.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Linear_Functional.py
new file mode 100644
index 0000000..6edc918
--- /dev/null
+++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Dual-Spaces/Linear_Functional.py
@@ -0,0 +1,29 @@
+from manimlib.imports import *

+import numpy as np

+class LinearFunctional(Scene):

+	def construct(self):

+		big_box=Rectangle().scale(2.7)		

+		box = Rectangle(height=2,width=2,color=DARK_GREY).set_fill(color=PURPLE,opacity=350)

+		arrow1 = Arrow(color=RED).shift(1.8*LEFT+0.5*UP)

+		arrow2 = Arrow(color=RED).shift(1.8*LEFT+0.5*DOWN)

+		arrow3 = Arrow(color=GREEN).shift(0.5*UP+1.8*RIGHT)

+		arrow4 = Arrow(color=GREEN).shift(0.5*DOWN+1.8*RIGHT)

+		Linear = TextMobject("LINEAR",color=BLACK).scale(0.5).shift(0.2*UP)

+		Functional = TextMobject("FUNCTIONAL",color=BLACK).scale(0.5).shift(0.2*DOWN)

+		u = TextMobject(r"$\vec{u}$",color=YELLOW).scale(0.7).shift(2.8*LEFT+0.5*UP)

+		v = TextMobject(r"$\vec{v}$",color=YELLOW).scale(0.7).shift(2.8*LEFT+0.5*DOWN)	

+		f1 = TextMobject(r"$f_1$",color=YELLOW).scale(0.7).shift(2.8*RIGHT+0.5*UP)

+		f2 = TextMobject(r"$f_2$",color=YELLOW).scale(0.7).shift(2.8*RIGHT+0.5*DOWN)		

+		text = TextMobject(r"The Linear Functional is a function that takes $\vec{u}, \vec{v} \in$ V as inputs and gives the output $f_1, f_2\in$ F.").scale(0.55).shift(2*DOWN)

+		self.play(ShowCreation(big_box))

+		self.play(ShowCreation(box))

+		self.play(ShowCreation(Linear),ShowCreation(Functional))

+		self.wait(2)

+		self.play(ShowCreation(arrow1),ShowCreation(u))

+		self.play(ShowCreation(arrow3),ShowCreation(f1))

+		self.wait(0.7)

+		self.play(ShowCreation(arrow2),ShowCreation(v))

+		self.play(ShowCreation(arrow4),ShowCreation(f2))		

+		self.wait(1)

+		self.play(ShowCreation(text))

+		self.wait(4)