### Instruction: Find the matrix $\mathbf{M}$ if it satisfies $\mathbf{M} \mathbf{i} = \begin{pmatrix} 2 \\ 3 \\ -8 \end{pmatrix},$ $\mathbf{M} \mathbf{j} = \begin{pmatrix} 0 \\ 5 \\ -2 \end{pmatrix},$ and $\mathbf{M} \mathbf{k} = \begin{pmatrix} 7 \\ -1 \\ 4 \end{pmatrix}.$.### Response: I need to find a 3 by 3 matrix $\mathbf{M}$ that maps the standard basis vectors $\mathbf{i}, \mathbf{j},$ and $\mathbf{k}$ to the given vectors.
One way to do that is to use the fact that the columns of a matrix are the images of the basis vectors under the matrix transformation.
That means I can write $\mathbf{M}$ as a matrix whose columns are the given vectors: $\mathbf{M} = \begin{pmatrix} 2 & 0 & 7 \\ 3 & 5 & -1 \\ -8 & -2 & 4 \end{pmatrix}.$
To check that this is correct, I can multiply $\mathbf{M}$ by each basis vector and see if I get the right result.
For example, $\mathbf{M} \mathbf{i} = \begin{pmatrix} 2 & 0 & 7 \\ 3 & 5 & -1 \\ -8 & -2 & 4 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \\ -8 \end{pmatrix},$ as expected.
Similarly, $\mathbf{M} \mathbf{j} = \begin{pmatrix} 0 \\ 5 \\ -2 \end{pmatrix}$ and $\mathbf{M} \mathbf{k} = \begin{pmatrix} 7 \\ -1 \\ 4 \end{pmatrix},$ as desired.
Therefore, I have found the correct matrix $\mathbf{M}.$