Finding a linear map when given a matrix

Question

I was wondering if anyone could help me. I am currently studying linear algebra and have came across some questions about finding a linear map associated to a given matrix.

I have been working a lot with the opposition of this statement with finding an associated matrix given the linear map but do not know how to tackle this new situation, although I have a few ideas.

So if I have a matrix $$E=\begin{pmatrix} 3 & -6\\ 1 & -2\end{pmatrix}$$

would the the linear map for this associated matrix be:

$$T=\{3 + x, -6 - 2x\}$$

or am I wrong?

Thanks for the help in advance xx

Answer 1

$$T[(x',y') \rightarrow (3x-6y, x-2y)]$$ Because $$\begin{pmatrix}x' \\ y' \end{pmatrix}= \begin{pmatrix} 3 & -6 \\ 1 & -2\end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$$

Answer 2

By matrix multiplication we obtain

$$\begin{pmatrix} 3 & -6\\ 1 & -2\end{pmatrix}\begin{pmatrix} x\\ y\end{pmatrix}=\begin{pmatrix} 3x-6y\\ x-2y\end{pmatrix}=(3x-6y, x-2y)$$