### Instruction: Find the matrix that corresponds to projecting onto the vector $\begin{pmatrix} 1 \\ 7 \end{pmatrix}.$.### Response: To project a vector $\mathbf{v}$ onto another vector $\mathbf{u}$, I can use the formula $\frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}$.
In matrix notation, this means multiplying $\mathbf{v}$ by the matrix $\frac{1}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u} \mathbf{u}^T$, where $\mathbf{u}^T$ is the transpose of $\mathbf{u}$.
In this case, $\mathbf{u} = \begin{pmatrix} 1 \\ 7 \end{pmatrix}$, so $\mathbf{u}^T = \begin{pmatrix} 1 & 7 \end{pmatrix}$ and $\mathbf{u} \cdot \mathbf{u} = 1^2 + 7^2 = 50$.
Therefore, the projection matrix is $\frac{1}{50} \begin{pmatrix} 1 \\ 7 \end{pmatrix} \begin{pmatrix} 1 & 7 \end{pmatrix} = \frac{1}{50} \begin{pmatrix} 1 & 7 \\ 7 & 49 \end{pmatrix}$.