T T T - linear transformation:
T ( u + v ) = T ( u ) + T ( v ) T(u+v)=T(u)+T(v) T ( u + v ) = T ( u ) + T ( v )
T ( k u ) = k T ( u ) T(ku)=kT(u) T ( k u ) = k T ( u )
1.
v = ( x 1 , x 2 ) ∈ R 2 v=(x_1,x_2) \in \mathbb{R} ^2 v = ( x 1 , x 2 ) ∈ R 2 - arbitrary vector.
It can be expressed as a linear combination of v 1 v_1 v 1 and v 2 v_2 v 2 :
v = a v 1 + b v 2 v=av_1+bv_2 v = a v 1 + b v 2
( x 1 , x 2 ) = a ( − 2 , 1 ) + b ( 1 , 3 ) = ( − 2 a + b , a + 3 b ) (x_1,x_2)=a(-2,1)+b(1,3)=(-2a+b,a+3b) ( x 1 , x 2 ) = a ( − 2 , 1 ) + b ( 1 , 3 ) = ( − 2 a + b , a + 3 b )
{ x 1 = − 2 a + b x 2 = a + 3 b { x 1 + 2 x 2 = 7 b x 2 = a + 3 b { b = x 1 7 + 2 x 2 7 a = − 3 x 1 7 + x 2 7 \begin{cases}
x_1=-2a+b \\ x_2=a+3b
\end{cases}
\; \;
\begin{cases}
x_1+2x_2=7b\\ x_2=a+3b
\end{cases}
\begin{cases}
b=\frac{x_1}{7}+\frac{2x_2}{7} \\a=\frac{-3x_1}{7}+\frac{x_2}{7}
\end{cases} { x 1 = − 2 a + b x 2 = a + 3 b { x 1 + 2 x 2 = 7 b x 2 = a + 3 b { b = 7 x 1 + 7 2 x 2 a = 7 − 3 x 1 + 7 x 2
T ( v ) = T ( a v 1 + b v 2 ) = a T ( v 1 ) + b T ( v 2 ) = a ( − 1 , 2 , 0 ) + b ( 0 , − 3 , 5 ) = T(v)=T(av_1+bv_2)=aT(v_1)+bT(v_2)=a(-1,2,0)+b(0,-3,5)= T ( v ) = T ( a v 1 + b v 2 ) = a T ( v 1 ) + b T ( v 2 ) = a ( − 1 , 2 , 0 ) + b ( 0 , − 3 , 5 ) =
= ( − 3 x 1 7 + x 2 7 ) ( − 1 , 2 , 0 ) + ( x 1 7 + 2 x 2 7 ) ( 0 , − 3 , 5 ) = =(\frac{-3x_1}{7}+\frac{x_2}{7})(-1,2,0)+(\frac{x_1}{7}+\frac{2x_2}{7} )(0,-3,5)= = ( 7 − 3 x 1 + 7 x 2 ) ( − 1 , 2 , 0 ) + ( 7 x 1 + 7 2 x 2 ) ( 0 , − 3 , 5 ) =
= ( 3 x 1 7 + − x 2 7 , − 9 x 1 7 + − 4 x 2 7 , 5 x 1 7 + 10 x 2 7 ) = 1 7 ( 3 x 1 − x 2 , − 9 x 1 − 4 x 2 , 5 x 1 + 10 x 2 ) =(\frac{3x_1}{7}+\frac{-x_2}{7} , \frac{-9x_1}{7}+\frac{-4x_2}{7} , \frac{5x_1}{7}+\frac{10x_2}{7})=
\frac{1}{7}(3x_1-x_2,-9x_1-4x_2,5x_1+10x_2) = ( 7 3 x 1 + 7 − x 2 , 7 − 9 x 1 + 7 − 4 x 2 , 7 5 x 1 + 7 10 x 2 ) = 7 1 ( 3 x 1 − x 2 , − 9 x 1 − 4 x 2 , 5 x 1 + 10 x 2 )
T ( 2 , − 3 ) = 1 7 ( 9 , − 6 , − 20 ) = ( 9 7 , − 6 7 , − 20 7 ) T(2,-3)=\frac{1}{7}(9,-6,-20)=(\frac{9}7,-\frac{6}7,-\frac{20}7) T ( 2 , − 3 ) = 7 1 ( 9 , − 6 , − 20 ) = ( 7 9 , − 7 6 , − 7 20 )
2.
v = ( x 1 , x 2 , x 3 ) ∈ R 3 v=(x_1,x_2,x_3)\in \mathbb{R}^3 v = ( x 1 , x 2 , x 3 ) ∈ R 3 - arbitrary vector
It can bee expressed as a linear combination of v 1 , v 2 , v 3 : v_1,v_2,v_3: v 1 , v 2 , v 3 :
v = a v 1 + b v 2 + c v 3 v=av_1+bv_2+cv_3 v = a v 1 + b v 2 + c v 3
( x 1 , x 2 , x 3 ) = a ( 1 , 1 , 1 ) + b ( 1 , 1 , 0 ) + c ( 1 , 0 , 0 ) = ( a + b + c , a + b , a ) (x_1,x_2,x_3)=a(1,1,1)+b(1,1,0)+c(1,0,0)=(a+b+c,a+b,a) ( x 1 , x 2 , x 3 ) = a ( 1 , 1 , 1 ) + b ( 1 , 1 , 0 ) + c ( 1 , 0 , 0 ) = ( a + b + c , a + b , a )
{ x 1 = a + b + c x 2 = a + b x 3 = a { a = x 3 b = x 2 − x 3 c = x 1 − x 2 \begin{cases}
x_1=a+b+c \\
x_2=a+b \\
x_3=a
\end{cases}
\begin{cases}
a=x_3 \\ b=x_2-x_3 \\ c= x_1-x_2
\end{cases} ⎩ ⎨ ⎧ x 1 = a + b + c x 2 = a + b x 3 = a ⎩ ⎨ ⎧ a = x 3 b = x 2 − x 3 c = x 1 − x 2
T ( v ) = T ( a v 1 + b v 2 + c v 3 ) = a T ( v 1 ) + b T ( v 2 ) + c T ( v 3 ) = T(v)=T(av_1+bv_2+cv_3)=aT(v_1) +bT(v_2)+cT(v_3)= T ( v ) = T ( a v 1 + b v 2 + c v 3 ) = a T ( v 1 ) + b T ( v 2 ) + c T ( v 3 ) =
= a ( 3 , − 1 , 6 ) + b ( 4 , 0 , 1 ) + c ( − 1 , 7 , 1 ) = =a(3,-1,6)+b(4,0,1)+c(-1,7,1)= = a ( 3 , − 1 , 6 ) + b ( 4 , 0 , 1 ) + c ( − 1 , 7 , 1 ) =
= x 3 ( 3 , − 1 , 6 ) + ( x 2 − x 3 ) ( 4 , 0 , 1 ) + ( x 1 − x 2 ) ( − 1 , 7 , 1 ) = = x_3(3,-1,6)+(x_2-x_3)(4,0,1)+(x_1-x_2)(-1,7,1)= = x 3 ( 3 , − 1 , 6 ) + ( x 2 − x 3 ) ( 4 , 0 , 1 ) + ( x 1 − x 2 ) ( − 1 , 7 , 1 ) =
= ( − x 1 + 5 x 2 − x 3 , 7 x 1 − 7 x 2 − x 3 , x 1 + 5 x 3 ) =(-x_1+5x_2-x_3,7x_1-7x_2-x_3, x_1+5x_3) = ( − x 1 + 5 x 2 − x 3 , 7 x 1 − 7 x 2 − x 3 , x 1 + 5 x 3 )
T ( 3 , 6 , − 1 ) = ( 28 , − 20 , − 2 ) T(3,6,-1)=(28,-20,-2) T ( 3 , 6 , − 1 ) = ( 28 , − 20 , − 2 )
3.
T ( 4 v 1 − 5 v 2 + 6 v 3 ) = 4 T ( v 1 ) − 5 T ( v 2 ) + 6 T ( v 3 ) = T(4v_1-5v_2+6v_3)=4T(v_1)-5T(v_2)+6T(v_3)= T ( 4 v 1 − 5 v 2 + 6 v 3 ) = 4 T ( v 1 ) − 5 T ( v 2 ) + 6 T ( v 3 ) =
= 4 ( 1 , − 1 , 2 ) − 5 ( 0 , 3 , 2 ) + 6 ( − 3 , 1 , 2 ) = ( − 14 , − 13 , 10 ) =4(1,-1,2)-5(0,3,2)+6(-3,1,2)=(-14,-13,10) = 4 ( 1 , − 1 , 2 ) − 5 ( 0 , 3 , 2 ) + 6 ( − 3 , 1 , 2 ) = ( − 14 , − 13 , 10 )
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