### Instruction: Find the $2 \times 2$ matrix $\mathbf{M}$ such that
\[\mathbf{M} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 3 \\ 0 \end{pmatrix}\]and
\[\mathbf{M} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ -7 \end{pmatrix}.\].### Response: I need to find the matrix $\mathbf{M}$ that transforms the standard basis vectors $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ into the given vectors $\begin{pmatrix} 3 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ -7 \end{pmatrix}$.
I know that the columns of a matrix are the images of the standard basis vectors under the linear transformation defined by that matrix.
So, I can write $\mathbf{M}$ as a matrix whose first column is $\begin{pmatrix} 3 \\ 0 \end{pmatrix}$ and whose second column is $\begin{pmatrix} 2 \\ -7 \end{pmatrix}$.
That means $\mathbf{M} = \begin{pmatrix} 3 & 2 \\ 0 & -7 \end{pmatrix}$.