Answer to Question #233332 in Linear Algebra for Diego

Question

T : R

3 → R

2

 defined by : 9 

T(x, y, z) = (x -y + z, -2x + 2y -2 z)

Answer 1

1. Let $u=(u_1, u_2, u_3)$ and $v=(v_1, v_2, v_3)$ be vectors in $R^3$ and $c$ and $d$ be scalars.

Consider

$$T(cu+dv)=T(cu_1+dv_1, cu_2+dv_2, cu_3+dv_3)$$

$$=(cu_1+dv_1-(cu_2+dv_2)+cu_3+dv_3,$$

$$-2(cu_1+dv_1)+2(cu_2+dv_2)-2(cu_3+dv_3))$$

$$=(c(u_1-u_2+u_3)+d(v_1-v_2+v_3),$$

$$c(-2u_1+2u_2-2u_3)+d(-2v_1+2v_2-2v_3))$$

$$=c(u_1-u_2+u_3, -2u_1+2u_2-2u_3)$$

$$+d(v_1-v_2+v_3, -2v_1+2v_2-2v_3)$$

$$=cT(u)+dT(v)$$

Therefore the transformation $T:R^3\to R^2,$ given by

$$T(x, y, z)=(x-y+z, -2x+2y-2z)$$

is linear.

2.

$$T(e_1)=T(1, 0,0)$$

$$=(1-0+0, -2+0-0)=(1, -2)$$

$$T(e_2)=T(0, 1,0)$$

$$=(0-1+0, -0+2-0)=(-1, 2)$$

$$T(e_3)=T(0, 0,1)$$

$$=(0-0+1, -0+0-2)=(1, -2)$$

So

$$A=\begin{pmatrix} 1 & -1 & 1 \\ -2 & 2 & -2 \end{pmatrix}$$

3.

$$T(0,0,0)=(0,0)$$