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diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Conjugate_Symmetry_and_Positivity_of_Inner_Product.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Conjugate_Symmetry_and_Positivity_of_Inner_Product.py
new file mode 100644
index 0000000..1d84842
--- /dev/null
+++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Conjugate_Symmetry_and_Positivity_of_Inner_Product.py
@@ -0,0 +1,141 @@
+from manimlib.imports import *

+VECTORS = [[0, 2],

+           [1, 1],

+           [0, -2],

+           [1, -1],

+           [2, -2],

+           [2, 2],

+           [2, 2],

+           [-2, -1],

+           [3, 4]]

+class Scene1(LinearTransformationScene):

+    CONFIG = {

+        "include_background_plane": True,

+        "include_foreground_plane": False,

+        "show_coordinates": True,

+        "show_basis_vectors": False,

+        "basis_vector_stroke_width": 3,

+    }

+    def construct(self):

+      text1 = TextMobject(r"\text{$u$}",r"\text{ = $0 + 2i$, }",r"\text{$v$}",r"\text{ = $1 + i$}").scale(0.6).shift(3.5*UP+4*LEFT)      

+      text1[0].set_color(YELLOW)

+      text1[2].set_color(RED)

+      text1.add_background_rectangle()

+      self.play(ShowCreation(text1))

+      text2 = TextMobject(r"\text{$\overline{u}$}",r"\text{ = $0 - 2i$, }",r"\text{$\overline{v}$}",r"\text{ = $1 - i$}").scale(0.6).shift(3*UP+4*LEFT)      

+      text2[0].set_color(YELLOW)

+      text2[2].set_color(RED)

+      text2.add_background_rectangle()

+      self.play(ShowCreation(text2))

+      self.wait(2)

+      v1 = self.add_vector(VECTORS[0],color = YELLOW, stroke_width = 3.5)

+      u = TextMobject(r"$u$",color=YELLOW).shift(0.3*LEFT+2*UP).scale(0.6)

+      self.play(ShowCreation(u))

+      v1b = self.add_vector(VECTORS[2],color = YELLOW, stroke_width = 3.5)

+      ub = TextMobject(r"$\overline{u}$",color=YELLOW).shift(2*DOWN+0.3*LEFT).scale(0.6)

+      self.play(ShowCreation(ub))

+      self.wait(2)

+      v2 = self.add_vector(VECTORS[1],color = RED, stroke_width = 3.5)

+      v = TextMobject(r"$v$",color=RED).shift(1.2*RIGHT+1*UP).scale(0.6)

+      self.play(ShowCreation(v))

+      v2b = self.add_vector(VECTORS[3],color = RED, stroke_width = 3.5)

+      vb = TextMobject(r"$\overline{v}$",color=RED).shift(1.2*RIGHT+1*DOWN).scale(0.6)

+      self.play(ShowCreation(vb))

+      text3 = TextMobject(r"\text{$<u, v>$}",r"\text{ = }",r"\text{$\overline{u}$",r"\text{$\cdot$}",r"\text{$v$}",r"\text{ = }",r"\text{$2 - 2i$}").shift(2.5*UP+3.7*LEFT).scale(0.6)

+      text3[0].set_color(BLUE)

+      text3[2].set_color(YELLOW)

+      text3[4].set_color(RED)

+      text3.add_background_rectangle()

+      self.play(ShowCreation(text3))

+      self.wait(2)

+      text4 = TextMobject(r"\text{$<\overline{u, v}>$",r"\text{ = }",r"\text{$\overline{u}$",r"\text{$\cdot$}",r"\text{$v$}",r"\text{ = }",r"\text{$2 + 2i$}").shift(2*UP+3.7*LEFT).scale(0.6)

+      text4[0].set_color(BLUE)

+      text4[2].set_color(YELLOW)

+      text4[4].set_color(RED)

+      text4.add_background_rectangle()

+      line = Line(stroke_width = 1.5).scale(0.33).shift(2.2*UP+3.54*LEFT)

+      self.play(ShowCreation(text4),ShowCreation(line))

+      self.wait(2)

+      self.play(FadeOut(v1),FadeOut(v1b),FadeOut(v2),FadeOut(v2b),FadeOut(u),FadeOut(ub),FadeOut(v),FadeOut(vb))

+      v3 = self.add_vector(VECTORS[4],color = BLUE, stroke_width = 3.5)

+      uv = TextMobject(r"$\overline{u}\cdot v$",color=BLUE).shift(2.4*RIGHT+2.1*DOWN).scale(0.6)

+      self.play(ShowCreation(uv))

+      v3b = self.add_vector(VECTORS[5],color = BLUE, stroke_width = 3.5)

+      uvb = TextMobject(r"$\overline{\overline{u}\cdot v}$",color=BLUE).shift(2.4*RIGHT+2.1*UP).scale(0.6)

+      self.play(ShowCreation(uvb))

+      self.wait(2)

+      text5 = TextMobject(r"\text{$<v, u>$}",r"\text{ = }",r"\text{$\overline{v}$",r"\text{$\cdot$}",r"\text{$u$}",r"\text{ = }",r"\text{$2 + 2i$}").shift(1.5*UP+3.7*LEFT).scale(0.6)

+      text5[0].set_color(MAROON_B)

+      text5[2].set_color(RED)

+      text5[4].set_color(YELLOW)

+      text5.add_background_rectangle()

+      self.play(ShowCreation(text5))

+      self.wait(2)

+      v4 = self.add_vector(VECTORS[5],color = MAROON_B, stroke_width = 3.5)

+      vu = TextMobject(r"$\overline{v}\cdot u$",color=MAROON_B).shift(1.3*RIGHT+2.1*UP).scale(0.6)

+      self.play(ShowCreation(vu))

+      self.play(FadeOut(uvb))

+      self.wait(2)

+      text6 = TextMobject(r"\text{$<\overline{u, v}>$",r"\text{ = }",r"\text{$<v, u>$").scale(0.6).shift(0.8*UP+4.2*LEFT)

+      text6[0].set_color(BLUE)

+      text6[2].set_color(MAROON_B)

+      text6.add_background_rectangle()

+      self.play(ShowCreation(text6))

+      rect = Rectangle(height = 0.7,width = 3.5)

+      rect.surround(text6)

+      self.play(ShowCreation(rect))

+      self.wait(3)

+      self.play(FadeOut(line),FadeOut(text1),FadeOut(text2),FadeOut(text3),FadeOut(text4),FadeOut(text5),FadeOut(text6),FadeOut(v4),FadeOut(vu),FadeOut(v3),FadeOut(uv),FadeOut(rect),FadeOut(v3b))

+

+      text7 = TextMobject(r"\text{$u$}",r"\text{ = $(1 + i) > 0$}").scale(0.6).shift(3.5*UP+4.5*LEFT)

+      text7[0].set_color(YELLOW)

+      text7.add_background_rectangle()

+      self.play(ShowCreation(text7))

+      v5 = self.add_vector(VECTORS[1],color = YELLOW, stroke_width = 3.5)

+      u = TextMobject(r"$u$",color=YELLOW).shift(1.2*RIGHT+1*UP).scale(0.6)

+      self.play(ShowCreation(u))

+      self.wait(1.5)

+      text8 = TextMobject(r"\text{$<u, u>$}",r"\text{ = $(0 + 2i) > 0$ }").scale(0.6).shift(2.7*UP+4*LEFT)      

+      text8[0].set_color(GREEN)

+      text8.add_background_rectangle()

+      rect1 = Rectangle(height = 0.55, width = 3.3)

+      rect1.surround(text8)

+      self.play(ShowCreation(text8),ShowCreation(rect1))

+      self.wait(2)

+      v6 = self.add_vector(VECTORS[0],color = GREEN, stroke_width = 3.5)

+      uu = TextMobject(r"$<u, u>$",color=GREEN).shift(0.8*LEFT+1.9*UP).scale(0.6)

+      self.play(ShowCreation(uu))

+      text9 = TextMobject(r"\text{$v$}",r"\text{ = $(-2 - i) < 0$}").scale(0.6).shift(1.5*UP+4.4*LEFT)

+      text9[0].set_color(RED)

+      text9.add_background_rectangle()

+      self.play(ShowCreation(text9))

+      self.wait(1.5)

+      v7 = self.add_vector(VECTORS[7],color = RED, stroke_width = 3.5)

+      v = TextMobject(r"$v$",color=RED).shift(2.2*LEFT+1*DOWN).scale(0.6)

+      self.play(ShowCreation(v))

+      self.wait(1.5)

+      text10 = TextMobject(r"\text{$<v, v>$}",r"\text{ = $(3 + 4i) > 0$ }").scale(0.6).shift(0.7*UP+4*LEFT)      

+      text10[0].set_color(BLUE)

+      text10.add_background_rectangle()

+      rect2 = Rectangle(height=0.55,width=3.3)

+      rect2.surround(text10)

+      self.play(ShowCreation(text10),ShowCreation(rect2))

+      self.wait(2)

+      v8 = self.add_vector(VECTORS[8],color = BLUE, stroke_width = 3.5)

+      vv = TextMobject(r"$<v, v>$",color=BLUE).shift(2.1*RIGHT+3.8*UP).scale(0.6)

+      self.play(ShowCreation(vv))

+      self.wait(4)

+

+

+

+

+

+

+      

+

+

+

+

+

+

+

diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Inner_Product_Space_Example.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Inner_Product_Space_Example.py
new file mode 100644
index 0000000..1f98bad
--- /dev/null
+++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Inner_Product_Space_Example.py
@@ -0,0 +1,175 @@
+from manimlib.imports import *

+from scipy import sin,cos

+class Inner_Product_Space_Example(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=True)

+        curve1 = self.get_graph(lambda x : sin(x), x_min=0,x_max=2.3,color=RED)

+        curve2 = self.get_graph(lambda x : x, x_min=0,x_max=2.3,color=DARK_BLUE)

+        curve3 = self.get_graph(lambda x : 1.4, x_min=0,x_max=2.3,color=GREEN)

+        text1 = TextMobject(r"$f(x)$").scale(0.5).shift(1.77*DOWN+0.55*RIGHT)

+        

+        text2 = TextMobject(r"$g(x)$").scale(0.5).shift(0.15*DOWN+0.55*RIGHT)

+        text3 = TextMobject(r"$h(x)$").scale(0.5).shift(1.03*DOWN+0.55*RIGHT)

+

+        self.play(ShowCreation(curve1),ShowCreation(text1))

+        self.wait(1)

+        self.play(ShowCreation(curve2),ShowCreation(text2))

+        self.wait(1)

+        self.play(ShowCreation(curve3),ShowCreation(text3))

+        self.wait(2)

+        curve4 = self.get_graph(lambda x : sin(x) + x, x_min=0,x_max=2.3,color=YELLOW)

+        text4 = TextMobject(r"$f(x) + g(x)$").scale(0.5).shift(0.6*UP+1*RIGHT)

+        self.wait(1.5)

+

+        self.play(ShowCreation(curve4),ShowCreation(text4),FadeOut(curve2),FadeOut(text2),FadeOut(curve1),FadeOut(text1))

+        self.wait(1.5)

+        text5 = TextMobject(r"\text{$<f(x) + g(x), h(x)>$ = ",r"\text{$\int_{a}^{b} (f(x) + g(x))h(x)$ $dx$}").scale(0.57).shift(4*RIGHT+3.5*UP)

+        text5[1].set_color(ORANGE)

+        self.play(ShowCreation(text5))

+

+        curve5 = self.get_graph(lambda x : (sin(x) + x)*1.6, x_min=0,x_max=2.3,color=ORANGE)

+        text6 = TextMobject(r"$(f(x) + g(x))\cdot h(x)$").scale(0.5).shift(2.2*UP+1.4*RIGHT)

+        area1 = self.get_area(curve5,0,2.3)

+        area1.set_color(ORANGE)

+        self.wait(1)

+        self.play(FadeOut(curve4),FadeOut(text4),FadeOut(curve3),FadeOut(text3),ShowCreation(curve5),ShowCreation(text6),ShowCreation(area1))

+        self.wait(2)

+        text7 = TextMobject(r"\text{$<f(x), h(x)>$ = ",r"\text{$\int_{a}^{b} (f(x)h(x)$ $dx$}").scale(0.57).shift(5*RIGHT+3*UP)

+        text7[1].set_color(BLUE)

+        self.play(ShowCreation(text7))

+        self.wait(1.5)

+        curve6 = self.get_graph(lambda x : (sin(x))*1.6, x_min=0,x_max=2.3,color=BLUE)

+        text8 = TextMobject(r"$f(x)\cdot h(x)$").scale(0.5).shift(1.4*DOWN+0.8*RIGHT)

+        area2 = self.get_area(curve6,0,2.3)

+        self.play(ShowCreation(curve6),ShowCreation(text8),ShowCreation(area2))

+        self.wait(1.5)

+        text9 = TextMobject(r"\text{$<g(x), h(x)>$ = ",r"\text{$\int_{a}^{b} (g(x)h(x)$ $dx$}").scale(0.57).shift(5*RIGHT+2.5*UP)

+        text9[1].set_color(MAROON_B)

+        self.play(ShowCreation(text9))

+        self.wait(1.5)

+        curve7 = self.get_graph(lambda x : x*1.6, x_min=0,x_max=2.3,color=MAROON_B)

+        text10 = TextMobject(r"$g(x)\cdot h(x)$").scale(0.5).shift(0.8*RIGHT+1*UP)

+        area3 = self.get_area(curve7,0,2.3)

+        area3.set_color(MAROON_B)

+        self.play(ShowCreation(curve7),ShowCreation(text10),ShowCreation(area3))

+        self.wait(2.6)

+        curve8 = self.get_graph(lambda x : (sin(x))*1.6 + x*1.6, x_min=0,x_max=2.3,color=RED_C)

+        area4 = self.get_area(curve8,0,2.3)

+        area4.set_color(RED_C)

+        text11 = TextMobject(r"$f(x)h(x) + g(x)h(x)$").scale(0.5).shift(2.2*UP + 1.4*RIGHT)

+        self.play(FadeOut(curve6),FadeOut(text8),FadeOut(curve7),FadeOut(text10),FadeOut(area2),FadeOut(area3),ShowCreation(curve8),ShowCreation(area4))

+        self.wait(1)

+        self.play(Transform(text6,text11))

+        self.wait(1.7)

+        text12 = TextMobject(r"$<f(x) + g(x), h(x)>$ = $<f(x), h(x)>$ + $<g(x), h(x)>$").scale(0.465).shift(0.7*UP+4*RIGHT)

+        rect1 = Rectangle(height=0.5)

+        rect1.surround(text12)

+        self.play(ShowCreation(text12),ShowCreation(rect1))

+        self.wait(3)

+        self.play(FadeOut(text6),FadeOut(text5),FadeOut(text7),FadeOut(text9),FadeOut(text12),FadeOut(rect1),FadeOut(curve8),FadeOut(area4),FadeOut(text11),FadeOut(curve5),FadeOut(area1))

+

+        curve2.set_color(ORANGE)

+        self.play(ShowCreation(curve1),ShowCreation(text1))

+        self.wait(1)

+        self.play(ShowCreation(curve2),ShowCreation(text2))

+        self.wait(2)

+        curve9 = self.get_graph(lambda x : 2*sin(x), x_min=0,x_max=2.3,color=GREEN)

+        text13 = TextMobject(r"$2f(x)$").scale(0.5).shift(1.1*DOWN+0.55*RIGHT)

+        self.play(Transform(curve1,curve9),Transform(text1,text13))

+        self.wait(1.5)

+

+        text14 = TextMobject(r"\text{$<2f(x), g(x)>$ = ",r"\text{$\int_{a}^{b} (2f(x))g(x)$ $dx$}").scale(0.57).shift(4*RIGHT+3.5*UP)

+        text14[1].set_color(YELLOW)

+        self.play(ShowCreation(text14))

+        self.wait(2.2)

+        curve10 = self.get_graph(lambda x : 2*sin(x)*x, x_min=0,x_max=2.3,color=YELLOW)

+        text15 = TextMobject(r"$2f(x)\cdot g(x)$").scale(0.5).shift(1*RIGHT+0.97*UP)

+        area5 = self.get_area(curve10,0,2.3)

+        area5.set_color(YELLOW)

+        self.play(ShowCreation(area5),ShowCreation(curve10),ShowCreation(text15),FadeOut(curve1),FadeOut(text1),FadeOut(curve2),FadeOut(text2))

+        self.wait(2)

+        text16 = TextMobject(r"\text{$<f(x), g(x)>$ = ",r"\text{$\int_{a}^{b} f(x)g(x)$ $dx$}").scale(0.57).shift(3.8*RIGHT+2.9*UP)

+        text16[1].set_color(TEAL)

+        self.play(ShowCreation(text16))

+        self.wait(1.7)

+        curve11 = self.get_graph(lambda x : sin(x)*x, x_min=0,x_max=2.3,color=TEAL)

+        area6 = self.get_area(curve11,0,2.3)

+        area6.set_color(TEAL)

+        text17 = TextMobject(r"$f(x)\cdot g(x)$").scale(0.5).shift(0.9*RIGHT+0.7*DOWN)

+        self.play(ShowCreation(curve11),ShowCreation(text17),ShowCreation(area6))

+        self.wait(2)

+

+        text18 = TextMobject(r"\text{$2 <f(x), g(x)>$ = ",r"\text{$2 \int_{a}^{b} f(x)g(x)$ $dx$}").scale(0.57).shift(4*RIGHT+2.3*UP)

+        text18[1].set_color(DARK_BLUE)

+        self.play(ShowCreation(text18))

+        self.wait(2)

+        curve12 = self.get_graph(lambda x : 2*sin(x)*x, x_min=0,x_max=2.3,color=DARK_BLUE)

+        area7 = self.get_area(curve12,0,2.3)

+        area7.set_color(DARK_BLUE)

+        text19 = TextMobject(r"= $2( f(x)\cdot g(x) )$").scale(0.5).shift(2.5*RIGHT+0.97*UP)

+        self.play(ShowCreation(curve12),ShowCreation(area7),ShowCreation(text19),FadeOut(text17),FadeOut(area6),FadeOut(curve11))

+

+        self.wait(2.5)

+        text20 = TextMobject(r"$<2f(x), g(x)>$ = $2<f(x), g(x)>$").scale(0.57).shift(0.6*DOWN+4*RIGHT)

+        rect2 = Rectangle(height=0.5)

+        rect2.surround(text20)

+        self.play(ShowCreation(text20),ShowCreation(rect2))

+        self.wait(3)

+

+        self.play(FadeOut(text14),FadeOut(text15),FadeOut(text19),FadeOut(text16),FadeOut(text18),FadeOut(rect2),FadeOut(curve10),FadeOut(area5),FadeOut(curve12),FadeOut(area7),FadeOut(text20))

+        curve1 = self.get_graph(lambda x : sin(x), x_min=0,x_max=2.3,color=YELLOW)

+        text1 = TextMobject(r"$f(x)$").scale(0.5).shift(1.77*DOWN+0.55*RIGHT)

+        self.play(ShowCreation(curve1),ShowCreation(text1))

+        self.wait(1.5)

+        self.play(ShowCreation(curve2),ShowCreation(text2))

+        self.wait(1.7)

+        text21 = TextMobject(r"\text{$<f(x), g(x)>$ = ",r"\text{$\int_{a}^{b} f(x)g(x)$ $dx$}").scale(0.57).shift(3.5*RIGHT+3*UP)

+        text21[1].set_color(GREEN)

+        self.play(ShowCreation(text21))

+        self.wait(2)

+        curve13 = self.get_graph(lambda x : sin(x)*x, x_min=0,x_max=2.3,color=GREEN)

+        area8 = self.get_area(curve13,0,2.3)

+        area8.set_color(GREEN)

+        text22 = TextMobject(r"$f(x)\cdot g(x)$").scale(0.5).shift(0.8*RIGHT+0.7*DOWN)

+        self.play(ShowCreation(curve13),ShowCreation(area8),ShowCreation(text22),FadeOut(curve1),FadeOut(text1),FadeOut(curve2),FadeOut(text2))

+        self.wait(2.2)

+        curve14 = self.get_graph(lambda x : sin(x)*x, x_min=0,x_max=2.3,color=RED)

+        area9 = self.get_area(curve14,0,2.3)

+        area9.set_color(RED)

+        self.play(ShowCreation(curve14),ShowCreation(area9))

+        text23 = TextMobject(r"= $\overline{f(x)\cdot g(x)}$").scale(0.5).shift(0.7*DOWN+2.1*RIGHT)

+        self.play(ShowCreation(text23))

+        self.wait(2)

+        text24 = TextMobject(r"For all the real functions").scale(0.5).shift(2*RIGHT+2*UP)

+        text25 = TextMobject(r"$<\overline{f(x), g(x)}>$ = $<f(x), g(x)>$").scale(0.5).shift(2*RIGHT+1.4*UP)

+        rect3 = Rectangle(height=0.7)

+        rect3.surround(text25)

+        self.play(ShowCreation(text24),ShowCreation(text25),ShowCreation(rect3))

+        self.wait(3)

+

+        

+

+

+

+

+

+

+

+

+

+

+

+

+

+

diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Inner_product.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Inner_product.py
new file mode 100644
index 0000000..e1344f7
--- /dev/null
+++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Inner_product.py
@@ -0,0 +1,28 @@
+from manimlib.imports import *

+import numpy as np

+class InnerProduct(Scene):

+	def construct(self):

+		big_box=Rectangle().scale(2.7)		

+		box = Rectangle(height=2,width=2,color=DARK_GREY).set_fill(color=BLUE_B,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(1.8*RIGHT)

+		Inner = TextMobject("INNER",color=BLACK).scale(0.65).shift(0.2*UP)

+		Product = TextMobject("PRODUCT",color=BLACK).scale(0.65).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)

+		scalar = TextMobject("(Scalar)").scale(0.55).shift(3.3*RIGHT+0.4*DOWN)

+		u.v = TextMobject(r"$<\vec{u}, \vec{v}>$",color=YELLOW).scale(0.7).shift(3.3*RIGHT)

+		V = TextMobject("V").scale(0.7).shift(2*UP+3.8*RIGHT)

+		text = TextMobject(r"The Inner Product is an operation that takes $\vec{u}, \vec{v} \in$ V as inputs and gives a scalar as an output.").scale(0.55).shift(2*DOWN)

+		self.play(ShowCreation(big_box),ShowCreation(V))

+		self.play(ShowCreation(box))

+		self.play(ShowCreation(Product),ShowCreation(Inner))

+		self.wait(2)

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

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

+		self.wait(1.5)

+		self.play(ShowCreation(arrow3),ShowCreation(scalar),ShowCreation(u.v))

+		self.wait(1)

+		self.play(ShowCreation(text))

+		self.wait(10)

diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Instances_of_Topological_Spaces.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Instances_of_Topological_Spaces.py
new file mode 100644
index 0000000..2b3cf14
--- /dev/null
+++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Instances_of_Topological_Spaces.py
@@ -0,0 +1,36 @@
+from manimlib.imports import *

+class InnerProduct(Scene):

+	def construct(self):

+		

+		circle1 = Circle(color=DARK_GREY).scale(3.7)

+		circle1.set_fill(color=GOLD,opacity=350)

+		square1 = Square(color=DARK_GREY).scale(2.5)

+		square1.set_fill(color=DARK_BLUE,opacity=350)

+		square2=Square(color=DARK_GREY).scale(1.6).shift(0.2*DOWN)

+		square2.rotate(np.pi/4).set_fill(color=YELLOW,opacity=350)

+		square3 = Square(color=DARK_GREY).scale(1).shift(0.3*DOWN)

+		square3.set_fill(color=BLACK,opacity=350)

+		circle2 = Circle(color=DARK_GREY).scale(0.5).shift(0.45*DOWN)

+		circle2.set_fill(color=MAROON_A,opacity=350)

+		text1 = TextMobject("Topological Spaces",color=BLACK).scale(0.5).shift(3*UP)

+		text2 = TextMobject("Metric Spaces",color=BLACK).scale(0.5).shift(2.2*UP)

+		text3 = TextMobject("Normed",color=BLACK).scale(0.44).shift(1.4*UP)

+		text4 = TextMobject("Vector Spaces",color=BLACK).scale(0.44).shift(1.17*UP)

+		text5 = TextMobject("Inner Product",color=WHITE).scale(0.37).shift(0.5*UP)		

+		text6 = TextMobject("Spaces",color=WHITE).scale(0.37).shift(0.27*UP)

+		text7 = TextMobject("Dot",color=BLACK).scale(0.37).shift(0.3*DOWN)

+		text8 = TextMobject("Product",color=BLACK).scale(0.32).shift(0.5*DOWN)

+

+

+	

+

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

+		self.wait(1.5)

+		self.play(ShowCreation(square1),ShowCreation(text2))

+		self.wait(1.5)

+		self.play(ShowCreation(square2),ShowCreation(text3),ShowCreation(text4))

+		self.wait(1.5)

+		self.play(ShowCreation(square3),ShowCreation(text5),ShowCreation(text6))

+		self.wait(1.5)

+		self.play(ShowCreation(circle2),ShowCreation(text7),ShowCreation(text8))

+		self.wait(4)

diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Interpretation_of_Norm_as_Length.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Interpretation_of_Norm_as_Length.py
new file mode 100644
index 0000000..23c568d
--- /dev/null
+++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Interpretation_of_Norm_as_Length.py
@@ -0,0 +1,42 @@
+from manimlib.imports import *

+import numpy as np

+class Interpretation_of_norm_as_length(GraphScene):

+    CONFIG = {

+        "x_min" : 0,

+        "x_max" : 5,

+        "y_min" : 0,

+        "y_max" : 5,

+        "y_tick_frequency" : 1, 

+        "x_tick_frequency" : 1, 

+        "axes_color":LIGHT_GRAY,

+        "x_axis_width": 5,

+        "y_axis_height":5,

+        "graph_origin" : ORIGIN + 2*DOWN + 2*LEFT,

+        "enclude_zero_label": False

+        

+    }   

+    def construct(self):

+        self.setup_axes(animate=False)

+        dot = Dot().scale(0.5)

+        self.play(ShowCreation(dot))

+        origin = TextMobject(r"(0, 0)").scale(0.5).shift(2.5*LEFT+2.2*DOWN)

+        line1 = self.get_graph(lambda x : x, x_min=0,x_max=2,color=WHITE)

+        line2 = Line().rotate(np.pi/2).shift(1*DOWN)

+        text1 = TextMobject(r"$(v_1, 0)$").scale(0.5).shift(2.2*DOWN+0.2*RIGHT)

+        text2 = TextMobject(r"$(0, v_2)$").scale(0.5).shift(2.5*LEFT)

+        text3 = TextMobject(r"$(v_1, v_2)$").scale(0.5).shift(0.5*RIGHT)

+        text4 = TextMobject(r"$| v_1 |$",color=RED_B).scale(0.5).shift(1*LEFT+2.3*DOWN)

+        text5 = TextMobject(r"$| v_2 |$",color=RED_B).scale(0.5).shift(0.3*RIGHT+1*DOWN)

+        text6 = TextMobject(r"$\sqrt{{v_1}^2 + {v_2}^2}$",color=RED_B).scale(0.5).rotate(np.pi/4).shift(1.3*LEFT+1*DOWN)

+        line3 = Line(color=YELLOW).shift(1*LEFT+2*DOWN)

+        self.play(ShowCreation(line1),ShowCreation(line2),ShowCreation(text1),ShowCreation(text2),ShowCreation(text3),ShowCreation(origin))

+        self.wait(1.5)

+        self.play(ShowCreation(line3),ShowCreation(text4))

+        self.wait(1.5)

+        line2 = Line(color=YELLOW).rotate(np.pi/2).shift(1*DOWN)

+        self.play(ShowCreation(line2),ShowCreation(text5))

+        self.wait(1.5)

+        line1 = self.get_graph(lambda x : x, x_min=0,x_max=2,color=YELLOW)

+        self.play(ShowCreation(text6),ShowCreation(line1))

+        self.wait(4)

+       
\ No newline at end of file
diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Linearity_of_Inner_Product.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Linearity_of_Inner_Product.py
new file mode 100644
index 0000000..e1344f7
--- /dev/null
+++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Linearity_of_Inner_Product.py
@@ -0,0 +1,28 @@
+from manimlib.imports import *

+import numpy as np

+class InnerProduct(Scene):

+	def construct(self):

+		big_box=Rectangle().scale(2.7)		

+		box = Rectangle(height=2,width=2,color=DARK_GREY).set_fill(color=BLUE_B,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(1.8*RIGHT)

+		Inner = TextMobject("INNER",color=BLACK).scale(0.65).shift(0.2*UP)

+		Product = TextMobject("PRODUCT",color=BLACK).scale(0.65).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)

+		scalar = TextMobject("(Scalar)").scale(0.55).shift(3.3*RIGHT+0.4*DOWN)

+		u.v = TextMobject(r"$<\vec{u}, \vec{v}>$",color=YELLOW).scale(0.7).shift(3.3*RIGHT)

+		V = TextMobject("V").scale(0.7).shift(2*UP+3.8*RIGHT)

+		text = TextMobject(r"The Inner Product is an operation that takes $\vec{u}, \vec{v} \in$ V as inputs and gives a scalar as an output.").scale(0.55).shift(2*DOWN)

+		self.play(ShowCreation(big_box),ShowCreation(V))

+		self.play(ShowCreation(box))

+		self.play(ShowCreation(Product),ShowCreation(Inner))

+		self.wait(2)

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

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

+		self.wait(1.5)

+		self.play(ShowCreation(arrow3),ShowCreation(scalar),ShowCreation(u.v))

+		self.wait(1)

+		self.play(ShowCreation(text))

+		self.wait(10)

diff --git a/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Vector_Projection.py b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Vector_Projection.py
new file mode 100644
index 0000000..b1724c1
--- /dev/null
+++ b/FSF-2020/linear-algebra/vector-spaces/Vector-Spaces/Inner-Product_Spaces/Vector_Projection.py
@@ -0,0 +1,43 @@
+from manimlib.imports import *

+VECTORS = [[3,1],

+           [1,3],

+           [1.96,0.65]]

+class Projection(LinearTransformationScene):

+    CONFIG = {

+        "include_background_plane": True,

+        "include_foreground_plane": False,

+        "show_coordinates": False,

+        "show_basis_vectors": False,

+        "basis_vector_stroke_width": 3,

+    }

+    def construct(self):

+      v1 = self.add_vector(VECTORS[0],color = YELLOW, stroke_width = 3.5)

+      u = TextMobject(r"$\vec{u}$",color=YELLOW).scale(0.65).shift(3.2*RIGHT,1*UP)

+      self.play(ShowCreation(u))

+      self.wait(0.7)

+      v2 = self.add_vector(VECTORS[1],color = RED, stroke_width = 3.5)

+      v = TextMobject(r"$\vec{v}$",color=RED).scale(0.65).shift(1.2*RIGHT,3*UP)

+      self.play(ShowCreation(v))

+      self.wait(0.7)

+      angle = Arc(radius=0.6).scale(0.5)

+      theta = TextMobject(r"$\theta$").scale(0.65).shift(0.5*UP+0.5*RIGHT).rotate(np.pi/6)

+      self.play(ShowCreation(angle),ShowCreation(theta))

+      self.wait(1)

+      line1 = Line().scale(1.25).rotate(np.pi/(2)).shift(1.8*UP+1.47*RIGHT)

+      line1.rotate(np.pi/8)     

+      self.play(ShowCreation(line1))

+      self.wait(1.3)

+      v3 = self.add_vector(VECTORS[2],color = BLUE, stroke_width = 3.5)

+      text1 = TextMobject(r"$\vec{v}$cos$\theta$",color=BLUE).scale(0.55).shift(1.25*RIGHT+0.26*UP).rotate(np.pi/9)

+      self.play(ShowCreation(text1))

+      self.wait(2)

+      text2 = TextMobject(r"\text{Projection on }",r"\text{$\vec{v}$}",r"\text{ onto }",r"\text{$\vec{u}$}",r"\text{ = }",r"\text{$ |\vec{v}|cos\theta$}").scale(0.6).shift(4*RIGHT+3*UP)

+      text2[1].set_color(RED)

+      text2[3].set_color(YELLOW)

+      text2[5].set_color(BLUE)

+      text2.add_background_rectangle()

+      self.play(ShowCreation(text2))

+      self.wait(4)

+      

+

+      
\ No newline at end of file