How to find the matrix A with the given linear transformation? I'm working on an assignment and I came across this problem and I am not really sure how to approach it. Any advice would be really helpful.
Find an example of a linear transformation $T:\mathbb{R}^2\to\mathbb{R}^3$ given by $T(x)=Ax$ such that
$T([1,1]) = \langle3,3,5\rangle$. Find the matrix $A$.
I know I have to use the standard basis, but I can't figure the process to find the matrix.

Why not let
$$A=\begin{bmatrix} 3 & 0 \\ 3 & 0 \\ 5 & 0 \end{bmatrix}$$
Then you'd have that
$$T(x)=Ax=\begin{bmatrix}  3 & 0 \\ 3 & 0 \\ 5 & 0  \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}=\begin{bmatrix} 3x_1 \\ 3x_1 \\ 5x_1 \end{bmatrix}.$$
And it would follow easily that $T\left(\begin{bmatrix}1\\1\end{bmatrix}\right)=\begin{bmatrix}3\\3\\5\end{bmatrix}$.
