### Instruction: Find the matrix that corresponds to reflecting over the vector $\begin{pmatrix} 3 \\ 2 \end{pmatrix}.$.### Response: To find the matrix that corresponds to reflecting over a vector, I need to recall the formula for the reflection matrix in terms of the vector's components.
The formula is $R = \frac{1}{a^2 + b^2} \begin{pmatrix} a^2 - b^2 & 2ab \\ 2ab & b^2 - a^2 \end{pmatrix}$, where $\begin{pmatrix} a \\ b \end{pmatrix}$ is the vector.
In this case, $a = 3$ and $b = 2$, so I just need to plug them in and simplify.
I get $R = \frac{1}{13} \begin{pmatrix} 5 & 12 \\ 12 & -5 \end{pmatrix}$.