Let $T$ be a linear transformation on the plane with
$T\begin{pmatrix}-5\\ -4\end{pmatrix}=\begin{pmatrix}-5\\ -5\end{pmatrix}$
and
$T\begin{pmatrix}-4\\ -3\end{pmatrix}=\begin{pmatrix}-1\\ 2\end{pmatrix}$
Find a standard matrix for $T(x)$.
Since the answer should be a $2$x$2$ matrix, I thought that I would solve for $T$ in the first equation and then solve for $T$ in the second equation. Then combine the two results into a standard matrix. But I am not sure how I would solve for $T$ as the inverse of a $2$x$1$ matrix is not possible.
So I am assuming there must be a different approach.
If so, any help or guidance will be highly appreciated!
Best Answer
Express the the standard basis in terms of $\begin{pmatrix}-5\\ -4\end{pmatrix}$ and $\begin{pmatrix}-4\\ -3\end{pmatrix}$ ,i.e., find $x_1$ and $x_2$ such that $\begin{pmatrix} 1\\ 0\end{pmatrix} = x_1\begin{pmatrix}-5\\ -5\end{pmatrix} + x_2\begin{pmatrix}-4\\ 3\end{pmatrix} $. Do the same for the basis $\begin{pmatrix} 0\\ 1\end{pmatrix} $. Now, compute $T(\begin{pmatrix} 0\\ 1\end{pmatrix})$ and $T(\begin{pmatrix} 1\\ 0\end{pmatrix})$ using the fact that $T$ is a linear transformation. From the values of $T(\begin{pmatrix} 0\\ 1\end{pmatrix})$ and $T(\begin{pmatrix} 1\\ 0\end{pmatrix})$, you can construct your matrix.