\[e_1=\begin{bmatrix}1 \\0 \end{bmatrix}\]

\[e_2=\begin{bmatrix}0 \\1 \end{bmatrix}\]

The task is given

\[T(e_1)=\begin{bmatrix}3 \\1 \\3 \\1 \end{bmatrix}\]

\[T(e_2)=\begin{bmatrix}-5 \\2 \\0 \\0 \end{bmatrix}\]

There exists a unique matrix A for the linear transformation T for which it holds \(\displaystyle{T}{\left({u}\right)}={A}{u}\) for all u and A is a form \(\displaystyle{A}={\left[{T}{\left({e}_{{1}}\right)},\ldots,{T}{\left({e}_{{n}}\right)}\right]}\), where \(\displaystyle{e}_{{i}},{i}={1},{2},\ldots\) are vectors from the identity matrix, respectively to columns.

Matrix A is the standart matrix for the linear transformation T.

So, the standart form is:

\[A=[T(e_1) T(e_2)]=\begin{bmatrix}3 & -5 \\1 & 2 \\3 & 0 \\1 & 0 \end{bmatrix}\]

\[A=\begin{bmatrix}3 & -5 \\1 & 2 \\3 & 0 \\1 & 0 \end{bmatrix}\]