Answer to Question #265228 in Linear Algebra for Jannat Butt

Question

Let

T:

R3→R2

be a linear transformation such that

(1,1,1) = (1,0),

(1,1,0) = (2, −1) and

(1,0,0) =

(4,3). What is

(2, −3,5)?

Answer 1

$$T\bigg(\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\bigg)=\begin{bmatrix} 1 \\ 0 \end{bmatrix},$$

$$T\bigg(\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}\bigg)=\begin{bmatrix} 2 \\ -1 \end{bmatrix},$$

$$T\bigg(\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}\bigg)=\begin{bmatrix} 6 \\ 3 \end{bmatrix}$$

$$A=\begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix}$$

$$\begin{matrix} a+b+c=1 \\ d+ e+f=0 \end{matrix}$$

$$\begin{matrix} a+b=2 \\ d+ e=-1 \end{matrix}$$

$$\begin{matrix} a=4 \\ d=3 \end{matrix}$$

$$A=\begin{bmatrix} 4 & -2 & -1 \\ 3 & -4 & 1 \end{bmatrix}$$

$$\begin{bmatrix} 4 & -2 & -1 \\ 3 & -4 & 1 \end{bmatrix}\begin{bmatrix} 2 \\ -3 \\ 5 \end{bmatrix}=\begin{bmatrix} 4(2)-2(-3)-1(5) \\ 3(2)-4(-3)+1(5) \end{bmatrix}$$

$$=\begin{bmatrix} 9 \\ 23 \end{bmatrix}$$