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diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Function_Scalar_Multiplication.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Function_Scalar_Multiplication.py
new file mode 100644
index 0000000..ac74792
--- /dev/null
+++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Function_Scalar_Multiplication.py
@@ -0,0 +1,62 @@
+from manimlib.imports import *

+from scipy import exp

+class FunctionScalarMultiplication(GraphScene):

+    CONFIG = {

+        "x_min" : -5,

+        "x_max" : 5,

+        "y_min" : -5,

+        "y_max" : 5,

+        "y_tick_frequency" : 1, 

+        "x_tick_frequency" : 1, 

+        "axes_color":LIGHT_GRAY,

+        "graph_origin": ORIGIN,

+        }

+    def construct(self):

+        self.setup_axes(animate = True)

+        curve1 = self.get_graph(lambda x : 1/3*(x**2)+1, x_min=-2,x_max=2.5,color=ORANGE)

+        curve2 = self.get_graph(lambda x : 2*(1/3*(x**2)+1), x_min=-2,x_max=2.5,color=GOLD)

+        fx= TextMobject("$f(x)$").scale(0.6).shift(1.25*UP + 2.2*LEFT)

+        gx= TextMobject("$2 \cdot f(x)$").scale(0.6).shift(0.5*UP + 2.1*LEFT)

+        f1 = self.get_vertical_line_to_graph(-2,curve1,color=YELLOW)

+        f2 = self.get_vertical_line_to_graph(-1.5,curve1,color=YELLOW)

+        f3 = self.get_vertical_line_to_graph(-1,curve1,color=YELLOW)

+        f4 = self.get_vertical_line_to_graph(-0.5,curve1,color=YELLOW)

+        f5 = self.get_vertical_line_to_graph(0,curve1,color=YELLOW)

+        f6 = self.get_vertical_line_to_graph(0.5,curve1,color=YELLOW)

+        f7 = self.get_vertical_line_to_graph(1,curve1,color=YELLOW)

+        f8 = self.get_vertical_line_to_graph(1.5,curve1,color=YELLOW)

+        f9 = self.get_vertical_line_to_graph(2,curve1,color=YELLOW)

+        f10 = self.get_vertical_line_to_graph(2.5,curve1,color=YELLOW)

+        self.play(ShowCreation(curve1),ShowCreation(fx))

+        self.wait(1.5)

+        self.play(ShowCreation(f1),ShowCreation(f2),ShowCreation(f3),ShowCreation(f4),ShowCreation(f5),ShowCreation(f6),ShowCreation(f7),ShowCreation(f8),ShowCreation(f9),ShowCreation(f10))

+        self.wait(1.7)

+        line1=Line(color=YELLOW).shift(5*LEFT+1.8*UP).scale(0.5)

+        line1.rotate(np.pi/2)

+        scalar = TextMobject("2 x").scale(0.65).shift(5.5*LEFT+1.9*UP)

+        equal = TextMobject("=").scale(0.65).shift(4.5*LEFT+1.9*UP)

+        line2=Line(color=BLUE).shift(4*LEFT+2.3*UP)

+        line2.rotate(np.pi/2)

+        self.play(ShowCreation(line1),ShowCreation(scalar),ShowCreation(equal),ShowCreation(line2))

+        g1 = self.get_vertical_line_to_graph(-2,curve2,color=BLUE)

+        g2 = self.get_vertical_line_to_graph(-1.5,curve2,color=BLUE)

+        g3 = self.get_vertical_line_to_graph(-1,curve2,color=BLUE)

+        g4 = self.get_vertical_line_to_graph(-0.5,curve2,color=BLUE)

+        g5 = self.get_vertical_line_to_graph(0,curve2,color=BLUE)

+        g6 = self.get_vertical_line_to_graph(0.5,curve2,color=BLUE)

+        g7 = self.get_vertical_line_to_graph(1,curve2,color=BLUE)

+        g8 = self.get_vertical_line_to_graph(1.5,curve2,color=BLUE)

+        g9 = self.get_vertical_line_to_graph(2,curve2,color=BLUE)

+        g10 = self.get_vertical_line_to_graph(2.5,curve2,color=BLUE)

+        self.play(ShowCreation(g1),ShowCreation(g2),ShowCreation(g3),ShowCreation(g4),ShowCreation(g5),ShowCreation(g6),ShowCreation(g7),ShowCreation(g8),ShowCreation(g9),ShowCreation(g10))

+        self.wait(2)

+        fx2=TextMobject("2$\cdot f(x)$").scale(0.6).shift(2.6*UP+2.3*LEFT)

+        self.play(ShowCreation(curve2),ShowCreation(fx2))

+        self.wait(1.5)

+        self.play(FadeOut(curve1),FadeOut(fx))

+        sc_mult=TextMobject("$(2 f(x)) = 2f(x)$",color=GOLD).scale(0.65).shift(0.65*UP + 5*LEFT)

+        rect = Rectangle(height=0.5,width=2)

+        rect.surround(sc_mult)

+        self.play(ShowCreation(sc_mult),ShowCreation(rect))

+        self.wait(3)

+

diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Function_Space_Addition.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Function_Space_Addition.py
new file mode 100644
index 0000000..5ce0e11
--- /dev/null
+++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Function_Space_Addition.py
@@ -0,0 +1,89 @@
+from manimlib.imports import *

+from scipy import sin,exp

+class FunctionalVectorSpace(GraphScene):

+    CONFIG = {

+        "x_min" : -5,

+        "x_max" : 5,

+        "y_min" : -5,

+        "y_max" : 5,

+        "y_tick_frequency" : 1, 

+        "x_tick_frequency" : 1, 

+        "axes_color":LIGHT_GRAY,

+        "graph_origin": ORIGIN,

+        

+

+    }

+    def construct(self):

+        self.setup_axes(animate = True)

+        curve1 = self.get_graph(lambda x : exp(x) + 0.5, x_min=-2,x_max=2.5,color=ORANGE)

+        curve2 = self.get_graph(lambda x : 1/3*(x**2)+1, x_min=-2,x_max=2.5,color=DARK_BLUE)

+        curve3 = self.get_graph(lambda x : 1/3*(x**2)+1 + exp(x) + 0.5, x_min=-2,x_max=2.5,color=BLACK)

+        fx= TextMobject("$f(x)$").scale(0.6).shift(1.25*UP + 2.1*LEFT)

+        gx= TextMobject("$g(x)$").scale(0.6).shift(0.5*UP + 2.1*LEFT)

+        f1 = self.get_vertical_line_to_graph(-2,curve1,color=YELLOW)

+        f2 = self.get_vertical_line_to_graph(-1.5,curve1,color=YELLOW)

+        f3 = self.get_vertical_line_to_graph(-1,curve1,color=YELLOW)

+        f4 = self.get_vertical_line_to_graph(-0.5,curve1,color=YELLOW)

+        f5 = self.get_vertical_line_to_graph(0,curve1,color=YELLOW)

+        f6 = self.get_vertical_line_to_graph(0.5,curve1,color=YELLOW)

+        f7 = self.get_vertical_line_to_graph(1,curve1,color=YELLOW)

+        f8 = self.get_vertical_line_to_graph(1.5,curve1,color=YELLOW)

+        f9 = self.get_vertical_line_to_graph(2,curve1,color=YELLOW)

+        f10 = self.get_vertical_line_to_graph(2.5,curve1,color=YELLOW)

+

+        self.play(ShowCreation(curve1),ShowCreation(gx))

+        self.wait(1.2)

+        self.play(ShowCreation(curve2),ShowCreation(fx))

+        self.wait(1.2)

+        self.play(ShowCreation(f1),ShowCreation(f2),ShowCreation(f3),ShowCreation(f4),ShowCreation(f5),ShowCreation(f6),ShowCreation(f7),ShowCreation(f8),ShowCreation(f9),ShowCreation(f10))

+        self.wait(1.7)

+        g1 = self.get_vertical_line_to_graph(-2,curve2,color=BLUE)

+        g2 = self.get_vertical_line_to_graph(-1.5,curve2,color=BLUE)

+        g3 = self.get_vertical_line_to_graph(-1,curve2,color=BLUE)

+        g4 = self.get_vertical_line_to_graph(-0.5,curve2,color=BLUE)

+        g5 = self.get_vertical_line_to_graph(0,curve2,color=BLUE)

+        g6 = self.get_vertical_line_to_graph(0.5,curve2,color=BLUE)

+        g7 = self.get_vertical_line_to_graph(1,curve2,color=BLUE)

+        g8 = self.get_vertical_line_to_graph(1.5,curve2,color=BLUE)

+        g9 = self.get_vertical_line_to_graph(2,curve2,color=BLUE)

+        g10 = self.get_vertical_line_to_graph(2.5,curve2,color=BLUE)

+        self.play(ShowCreation(g1),ShowCreation(g2),ShowCreation(g3),ShowCreation(g4),ShowCreation(g5),ShowCreation(g6),ShowCreation(g7),ShowCreation(g8),ShowCreation(g9),ShowCreation(g10))

+        line1=Line(color=BLUE).shift(5*LEFT+2.3*UP).scale(0.25)

+        line1.rotate(np.pi/2)

+        line2=Line(color=YELLOW).shift(6*LEFT+2.5*UP).scale(0.5)

+        line2.rotate(np.pi/2)

+        line3=Line(color=PURPLE_B).shift(4*LEFT+2.7*UP).scale(0.75)

+        line3.rotate(np.pi/2)

+        add=TextMobject("+").shift(2.4*UP+5.5*LEFT).scale(0.7)

+        equal=TextMobject("=").shift(2.4*UP+4.5*LEFT).scale(0.7)

+        self.play(ShowCreation(line2),ShowCreation(line1),ShowCreation(add),ShowCreation(equal),ShowCreation(line3))

+        self.wait(2)

+        self.play(FadeOut(curve1),FadeOut(curve2))

+        self.wait(3)

+        h1 = self.get_vertical_line_to_graph(-2,curve3,color=PURPLE_B)

+        h2 = self.get_vertical_line_to_graph(-1.5,curve3,color=PURPLE_B)

+        h3 = self.get_vertical_line_to_graph(-1,curve3,color=PURPLE_B)

+        h4 = self.get_vertical_line_to_graph(-0.5,curve3,color=PURPLE_B)

+        h5 = self.get_vertical_line_to_graph(0,curve3,color=PURPLE_B)

+        h6 = self.get_vertical_line_to_graph(0.5,curve3,color=PURPLE_B)

+        h7 = self.get_vertical_line_to_graph(1,curve3,color=PURPLE_B)

+        h8 = self.get_vertical_line_to_graph(1.5,curve3,color=PURPLE_B)

+        h9 = self.get_vertical_line_to_graph(2,curve3,color=PURPLE_B)

+        h10 = self.get_vertical_line_to_graph(2.5,curve3,color=PURPLE_B)

+        

+        line1.shift(1*LEFT+0.9*UP)

+        equal.shift(0.3*LEFT+0.2*UP)

+        f=TextMobject("$f(x)$").scale(0.5).shift(5.6*LEFT+3.2*UP)

+        g=TextMobject("$g(x)$").scale(0.5).shift(5.6*LEFT+2.4*UP)

+        fg=TextMobject("$(f + g)(x)$").scale(0.5).shift(2.85*UP+3.3*LEFT)

+        self.play(FadeOut(add),ShowCreation(equal),ShowCreation(line1),ShowCreation(f),ShowCreation(g),ShowCreation(fg),FadeOut(fx),FadeOut(gx))

+        self.wait(1.7)

+        self.play(ShowCreation(h1),ShowCreation(h2),ShowCreation(h3),ShowCreation(h4),ShowCreation(h5),ShowCreation(h6),ShowCreation(h7),ShowCreation(h8),ShowCreation(h9),ShowCreation(h10))

+        curve3 = self.get_graph(lambda x : 1/3*(x**2)+1 + exp(x) + 0.5, x_min=-2,x_max=2.5,color=RED_A)

+        fgx=TextMobject("$(f + g)(x)$").scale(0.5).shift(1.65*UP+2.4*LEFT)

+        self.play(ShowCreation(curve3),ShowCreation(fgx))

+        sum=TextMobject("$(f + g)(x) = f(x) + g(x)$",color=GOLD).scale(0.65).shift(0.8*UP + 5*LEFT)

+        rect = Rectangle(height=0.5,width=2)

+        rect.surround(sum)

+        self.play(ShowCreation(sum),ShowCreation(rect))

+        self.wait(3)
\ No newline at end of file
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Integral_Properties.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Integral_Properties.py
new file mode 100644
index 0000000..97c0e09
--- /dev/null
+++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Function-and-Polynomial-Spaces/Integral_Properties.py
@@ -0,0 +1,77 @@
+from manimlib.imports import *

+from scipy import sin 

+class Integral_Properties(GraphScene):

+    CONFIG = {

+        "x_min" : 0,

+        "x_max" : 5,

+        "y_min" : 0,

+        "y_max" : 6,

+        "y_tick_frequency" : 1, 

+        "x_tick_frequency" : 1, 

+        "axes_color":LIGHT_GRAY,

+        "x_labeled_nums" : list(range(6)),

+        "y_labeled_nums" : list(range(6))

+    }   

+    def construct(self):

+        self.setup_axes(animate=False)

+        curve1 = self.get_graph(lambda x : sin(x), x_min=0,x_max=2.5,color=RED)

+        curve2 = self.get_graph(lambda x : x, x_min=0,x_max=2.5,color=DARK_BLUE)

+        curve3 = self.get_graph(lambda x : sin(x) + x, x_min=0,x_max=2.5,color=GREEN)

+        fx = TextMobject(r"$f(x)$").scale(0.5).shift(1*RIGHT+1.8*DOWN)

+        gx = TextMobject(r"$g(x)$").scale(0.5).shift(1*RIGHT)

+        sum = TextMobject(r"$f(x) + g(x)$").scale(0.5).shift(1.3*RIGHT+0.6*UP)

+        area1 = self.get_area(curve1,0,2.5)

+        area2 = self.get_area(curve2,0,2.5)

+        area3 = self.get_area(curve3,0,2.5)        

+        area2.set_fill(color=PURPLE)

+        area3.set_fill(color=ORANGE)

+        text1=TextMobject(r"$\int_{0}^{2.5}$ f(x) dx = Area under the curve f(x)",color=BLUE_C).scale(0.7).shift(2.7*RIGHT+3*UP)

+        text2=TextMobject(r"$\int_{0}^{2.5}$ g(x) dx = Area under the curve g(x)",color=PURPLE_B).scale(0.7).shift(2.7*RIGHT+2.4*UP)

+        text3=TextMobject(r"Area under the curve f(x) + g(x) = $\int_{0}^{2.5} (f(x) + g(x)) dx$",color=ORANGE).scale(0.7).shift(2.7*RIGHT+1.8*UP)

+        text4=TextMobject(r"\text{$\int_{0}^{2.5}$ (f(x) + g(x)) dx}",r"\text{ = }",r"\text{ $\int_{0}^{2.5}$ f(x) dx }",r"\text{+}",r"\text{$\int_{0}^{2.5}$ g(x) dx}").scale(0.62).shift(2.7*RIGHT+2.7*UP)

+        text4[0].set_color(ORANGE)

+        text4[2].set_color(BLUE_C)

+        text4[4].set_color(PURPLE_B)

+        self.play(ShowCreation(curve1), ShowCreation(fx))

+        self.wait(1.2)

+        self.play(ShowCreation(curve2),ShowCreation(gx))

+        self.wait(1.2)

+        self.play(ShowCreation(area1))

+        self.play(ShowCreation(text1))

+        self.wait(1.5)

+        self.play(ShowCreation(area2))

+        self.play(ShowCreation(text2))

+        self.wait(1.5)

+        self.play(ShowCreation(curve3),ShowCreation(sum))

+        self.play(ShowCreation(area3))

+        self.play(ShowCreation(text3))

+        self.wait(2)

+        self.play(FadeOut(text1),FadeOut(text2),FadeOut(text3))

+        self.wait(1)

+        self.play(ShowCreation(text4))

+        self.wait(3)

+        self.play(FadeOut(curve1),FadeOut(curve2),FadeOut(area1),FadeOut(area2))

+        self.wait(1.5)

+        self.play(FadeOut(text4),FadeOut(area2),FadeOut(curve2),FadeOut(gx),FadeOut(curve3),FadeOut(sum),FadeOut(area3),ShowCreation(curve1),ShowCreation(fx))

+        self.wait(1.5)

+        self.play(ShowCreation(area1),ShowCreation(text1))

+        self.wait(1.5)

+        curve4 = self.get_graph(lambda x : 2*sin(x), x_min=0,x_max=2.5,color=RED)

+        area4 = self.get_area(curve4,0,2.5)

+        area4.set_fill(color=YELLOW)

+        fx2 = TextMobject(r"$2f(x)$").scale(0.7).shift(1*RIGHT+1.2*DOWN)

+        scalar_mul=TextMobject(r"$\int_{0}^{2.5} ( 2f(x) ) dx$ = 2 $\times$ Area under the curve f(x)",color=YELLOW).scale(0.7).shift(2.7*RIGHT+2.4*UP)

+        self.play(ShowCreation(curve4),ShowCreation(fx2))

+        self.wait(1)

+        self.play(ShowCreation(area4))

+        self.wait(2)

+        self.play(ShowCreation(scalar_mul))

+        self.wait(2)

+        text5=TextMobject(r"\text{$\int_{0}^{2.5}$ (2 f(x)) dx}",r"\text{ = }",r"\text{2 $\int_{0}^{2.5}$ f(x) dx }").scale(0.67).shift(2.7*RIGHT+2.7*UP)

+        text5[0].set_color(YELLOW)

+        text5[2].set_color(BLUE_C)

+        self.play(FadeOut(text1),FadeOut(scalar_mul))

+        self.wait(1)

+        self.play(ShowCreation(text5))

+        self.wait(3)

+