### Instruction: Find the matrix that corresponds to projecting onto the vector $\begin{pmatrix} 2 \\ -3 \end{pmatrix}.$.### Response: To project onto a vector, I need to use the formula $P = \frac{vv^T}{v^Tv}$, where $v$ is the vector I want to project onto and $P$ is the projection matrix.
In this case, $v = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$, so I need to find $v^T$, the transpose of $v$, and $v^Tv$, the dot product of $v$ with itself.
The transpose of $v$ is just $v^T = \begin{pmatrix} 2 & -3 \end{pmatrix}$, a row vector.
The dot product of $v$ with itself is just $v^Tv = 2^2 + (-3)^2 = 4 + 9 = 13$.
Now I can plug these into the formula for $P$ and simplify: $P = \frac{vv^T}{v^Tv} = \frac{\begin{pmatrix} 2 \\ -3 \end{pmatrix}\begin{pmatrix} 2 & -3 \end{pmatrix}}{13} = \frac{1}{13}\begin{pmatrix} 4 & -6 \\ -6 & 9 \end{pmatrix}$.