How to find matrix of linear transformation given a system of four equations?

Question

Find the matrix of the linear transformation:

$y_1 = 9x_1+3x_2-3x_3$

$y_2 = 2x_1-9x_2+x_3$

$y_3 = 4x_1-9x_2-2x_3$

$y_4 = 5x_1+x_2+5x_3$

Answer 1

Firstly note that since $$T:\mathbb{R^3}\to\mathbb{R^4}$$

we are looking for a 4-by-3 matrix.

By definition, to find the first column set

$$C_1: \quad(x_1,x_2,x_3)=(1,0,0) \implies (y_1,y_2,y_3,y_4)=(9,2,4,5)\implies C_1=(9,2,4,5)^T$$

and so on fo the others

$$C_2: \quad(x_1,x_2,x_3)=(0,1,0)\implies (y_1,y_2,y_3,y_4)=(3,-9,-9,1)\implies C_2=(3,-9,-9,1)^T$$

$$C_3: \quad(x_1,x_2,x_3)=(0,0,1)\implies (y_1,y_2,y_3,y_4)=(-3,1,-2,5)\implies C_3=(-3,1,-2,5)^T$$

therefore

$$A=\begin{pmatrix} 9 & 3& -3 \\ 2 & -9&1\\4&-9&-2\\5&1&5 \end{pmatrix}$$