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from manimlib.imports import *
class Row_Space(Scene):
def construct(self):
Heading = TextMobject("Row Space")
defn1 = TextMobject("Definition 1: Row Space of a matrix is the linear combination of the rows of that matrix.")
defn2 = TextMobject("Definition 2: It is a vector space generated by a linear combination of the columns of $A^{T}$.")
equivalent = TextMobject(r"Definition 1 $\equiv$ Definition 2")
Heading.move_to(2*UP)
Heading.set_color(color = DARK_BLUE)
defn1.move_to(UP)
defn1.scale(0.75)
defn2.scale(0.75)
equivalent.move_to(DOWN)
self.play(Write(Heading))
self.play(Write(defn1))
self.play(Write(defn2))
self.play(Write(equivalent))
self.wait(2)
self.play(FadeOut(Heading),FadeOut(defn1),FadeOut(defn2),ApplyMethod(equivalent.move_to,2*UP))
how = TextMobject("Let us see, How?")
how.move_to(UP)
self.play(Write(how))
self.play(FadeOut(equivalent),FadeOut(how))
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.move_to(2*UP+3*LEFT)
A[1].set_color(color = DARK_BLUE)
A.scale(0.80)
self.play(Write(A))
rows = TextMobject(r"Rows of A $\rightarrow$",
r"$\left( \begin{array}{c c c} 1 & 2 & 1 \end{array} \right)$,",
r"$ \left( \begin{array}{c c c} 1 & 3 & 1 \end{array} \right)$,",
r"$\left( \begin{array}{c c c} 2 & 1 & 4 \end{array} \right)$,",
r"$ \left( \begin{array}{c c c} 3 & 2 & 3 \end{array} \right)$")
rows.scale(0.75)
rows[1:5].set_color(DARK_BLUE)
self.play(Write(rows))
ac_defn1 = TextMobject("According to Definition 1 : ")
ac_defn1.move_to(DOWN)
RS_A = TextMobject(r"Row Space of $A = x_{1}$",
r"$\left( \begin{array}{c c c} 1 & 2 & 1 \end{array} \right)$",
r"$+x_{2}$",
r"$ \left( \begin{array}{c c c} 1 & 3 & 1 \end{array} \right)$",
r"$ + x_{3}$",
r"$\left( \begin{array}{c c c} 2 & 1 & 4 \end{array} \right)$",
r"$+x_{4}$",
r"$ \left( \begin{array}{c c c} 3 & 2 & 3 \end{array} \right)$")
RS_A.move_to(DOWN+DOWN)
RS_A[6].move_to(2*DOWN+DOWN)
RS_A[7].move_to(2*DOWN+2*RIGHT+DOWN)
RS_A[1].set_color(color = DARK_BLUE)
RS_A[3].set_color(color = DARK_BLUE)
RS_A[5].set_color(color = DARK_BLUE)
RS_A[7].set_color(color = DARK_BLUE)
RS_A.scale(0.75)
self.play(FadeOut(rows[0]),Transform(rows[1],RS_A[1]),Transform(rows[2],RS_A[3]),Transform(rows[3],RS_A[5]),Transform(rows[4],RS_A[7]))
self.play(FadeIn(ac_defn1), Write(RS_A))
self.wait(1)
self.play(FadeOut(rows[1]), FadeOut(rows[2]), FadeOut(rows[3]), FadeOut(rows[4]), FadeOut(RS_A), FadeOut(ac_defn1))
A_T = TextMobject(r"$A^{T} = $",r"$\left( \begin{array}{c c c c} 1 & 1 & 2 & 3 \\ 2 & 3 & 1 & 2 \\ 1 & 1 & 4 & 3 \end{array} \right)$")
A_T.move_to(2*UP+3*RIGHT)
A_T[1].set_color(color = DARK_BLUE)
A_T.scale(0.80)
self.play(Write(A_T))
change1 = TextMobject(r"Rows of $A\equiv$ Columns of $A^{T}$")
change2 = TextMobject(r"Columns of $A\equiv$ Rows of $A^{T}$")
change2.move_to(DOWN)
change3 = TextMobject(r"Row Space of $A$ = Linear Combination of",r"Rows","of",r"A")
change3.move_to(2*DOWN)
change3[1].set_color(DARK_BLUE)
change3[3].set_color(DARK_BLUE)
self.play(Write(change1))
self.play(Write(change2))
self.play(Write(change3))
columns = TextMobject("Columns")
columns.scale(0.6)
columns.set_color(DARK_BLUE)
columns.move_to(2*DOWN+4.1*RIGHT)
a = TextMobject(r"$A^{T}$")
a.set_color(DARK_BLUE)
a.move_to(1.95*DOWN+5.6*RIGHT)
self.wait(0.5)
self.play(Transform(change3[1],columns), Transform(change3[3],a))
equal = TextMobject(r"= Column Space($A^{T}$)")
equal.move_to(3*DOWN+0.5*RIGHT)
self.play(Write(equal))
self.play(FadeOut(A_T), FadeOut(change1), FadeOut(change2), FadeOut(change3), FadeOut(A), FadeOut(equal))
ac_defn1.move_to(3*UP)
RS_A.move_to(1.5*UP)
RS_A[6].move_to(UP)
RS_A[7].move_to(UP+1.5*RIGHT)
self.play(Write(RS_A),FadeIn(ac_defn1))
CS_AT = TextMobject(r"Row Space of $A = x_{1}$",
r"$\left( \begin{array}{c} 1 \\ 2 \\ 1 \end{array} \right)$",
r"$+x_{2}$",
r"$ \left( \begin{array}{c} 1 \\ 3 \\ 1 \end{array} \right)$",
r"$ + x_{3}$",
r"$\left( \begin{array}{c} 2 \\ 1 \\ 4 \end{array} \right)$",
r"$+x_{4}$",
r"$ \left( \begin{array}{c} 3 \\ 2 \\ 3 \end{array} \right)$")
CS_AT.move_to(1.5*DOWN)
CS_AT[1].set_color(color = DARK_BLUE)
CS_AT[3].set_color(color = DARK_BLUE)
CS_AT[5].set_color(color = DARK_BLUE)
CS_AT[7].set_color(color = DARK_BLUE)
CS_AT.scale(0.75)
ac_defn2 = TextMobject("According to Definition 2 : ")
equivalent = TextMobject(r"Hence, Definition 1 $\equiv$ Definition 2")
equivalent.move_to(3*DOWN)
self.play(Write(CS_AT),FadeIn(ac_defn2))
self.play(Write(equivalent))
self.wait()
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