can someone help me with this Linear Algebra question:
Consider the linear transformation $f:R^2\rightarrow R^4$ given by: $$ f\begin{bmatrix}1\\3\end{bmatrix}=\begin{bmatrix}-2\\3\\14\\1\end{bmatrix}, f\begin{bmatrix}-1\\3\end{bmatrix}=\begin{bmatrix}-4\\-3\\10\\5\end{bmatrix} $$
What is the representative matrix for f?
Hint
Consider $e_1=\begin{bmatrix}1\\0\end{bmatrix}$, $e_2=\begin{bmatrix}0\\1\end{bmatrix}$ the canonical basis. Then:
$$f\begin{bmatrix}1\\3\end{bmatrix}=1\cdot f(e_1)+3\cdot f(e_2)=\begin{bmatrix}-2\\3\\14\\1\end{bmatrix}$$
And
$$f\begin{bmatrix}-1\\3\end{bmatrix}=-1\cdot f(e_1)+3\cdot f(e_2)=\begin{bmatrix}-4\\-3\\10\\5\end{bmatrix}$$
And once $e_1$ and $e_2$ are the canonical basis then $f(e_1)$ and $f(e_2)$ are the columns of the matrix.
What you have to do is only solve the above system for $f(e_1)$ and $f(e_2)$.
Can you finish?
Matrix $$ \begin{bmatrix}-2 & -4\\ 3 & -3\\14 & 10\\ 1 & 5 \\\end{bmatrix} $$ is from base $B = \lbrace (1, 3), (-1, 3) \rbrace$ to standard base. You can take matrix from standard bese to $B$ and multiple, receiving your matrix. It works like that(let $S$ be standard base and $f$ your function): $$ f: B \to S, g: S \to B $$ when you multiple you receive: $$ h = f \circ g : S \to S $$
And it's what you want.