### Instruction: Find the matrix that corresponds to reflecting over the vector $\begin{pmatrix} -1 \\ 3 \end{pmatrix}.$.### Response: To reflect over a vector, I need to find a matrix that preserves the length and angle of the vector, but flips the direction of any vector orthogonal to it.
A general matrix that does this is of the form $A = \frac{1}{a^2 + b^2}\begin{pmatrix} a^2 - b^2 & 2ab \\ 2ab & b^2 - a^2 \end{pmatrix}$, where $a$ and $b$ are the components of the vector.
In this case, $a = -1$ and $b = 3$, so I plug them in and simplify.
$A = \frac{1}{(-1)^2 + 3^2}\begin{pmatrix} (-1)^2 - 3^2 & 2(-1)(3) \\ 2(-1)(3) & 3^2 - (-1)^2 \end{pmatrix}$