### Instruction: Find the matrix that corresponds to projecting onto the vector $\begin{pmatrix} 1 \\ 7 \end{pmatrix}.$.### Response: To find the projection matrix, I need to use the formula $P = \frac{uu^T}{u^Tu}$, where $u$ is the vector I'm projecting onto.
In this case, $u = \begin{pmatrix} 1 \\ 7 \end{pmatrix}$, so I need to compute $uu^T$ and $u^Tu$.
First, $uu^T = \begin{pmatrix} 1 \\ 7 \end{pmatrix} \begin{pmatrix} 1 & 7 \end{pmatrix} = \begin{pmatrix} 1 & 7 \\ 7 & 49 \end{pmatrix}.$
Next, $u^Tu = \begin{pmatrix} 1 & 7 \end{pmatrix} \begin{pmatrix} 1 \\ 7 \end{pmatrix} = 1 + 49 = 50.$
Therefore, $P = \frac{uu^T}{u^Tu} = \frac{1}{50} \begin{pmatrix} 1 & 7 \\ 7 & 49 \end{pmatrix}.$