Find a matrix A such that T(x) = Ax given two linear transformations

Question

Suppose $T([a,-b])=[−x,y]$ and $T([a,b])=[x,y]$. Find a matrix $A$ such that $T(x)=Ax$ for all $x\in\mathbb{R}^2$.

Answer 1

Here's how you can turn this into a system of equations. Suppose you had the required matrix $ A=\begin{bmatrix} a & b \\ c & d \end{bmatrix} $

Then

$\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} 1 \\ -1 \end{bmatrix} =\begin{bmatrix} -1 \\ 1 \end{bmatrix} $ and $\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} 1 \\ 2 \end{bmatrix} =\begin{bmatrix} 2 \\ 4 \end{bmatrix} $

Multiply this out and set each component equal to eachother to get a system of 4 variables and 4 equations. Solve it then substitute back in $a,b,c,d$ to the matrix for the answer.

Answer 2

Hint: Write $e_1$ and $e_2$ in terms of $v_1=e_1-e_2$ and $v_2=e_1+2e_2$. Then use that $Tv_1=-v_1$ and $Tv_2=2v_2$.