### Instruction: Let $\mathbf{v}_0$ be a vector.  The vector $\mathbf{v}_0$ is projected onto $\begin{pmatrix} 3 \\ 1 \end{pmatrix},$ resulting in the vector $\mathbf{v}_1.$  The vector $\mathbf{v}_1$ is then projected onto $\begin{pmatrix} 1 \\ 1 \end{pmatrix},$ resulting in the vector $\mathbf{v}_2.$  Find the matrix that takes $\mathbf{v}_0$ to $\mathbf{v}_2.$.### Response: To find the matrix that takes $\mathbf{v}_0$ to $\mathbf{v}_2,$ I need to find the composition of the two projection matrices.
I recall that the projection matrix for a vector $\mathbf{u}$ is given by $\frac{1}{\|\mathbf{u}\|^2}\mathbf{u}\mathbf{u}^T,$ where $\mathbf{u}^T$ is the transpose of $\mathbf{u}.$
I apply this formula to the vectors $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$ to find their respective projection matrices.
For $\begin{pmatrix} 3 \\ 1 \end{pmatrix},$ I have $\|\mathbf{u}\|^2 = 3^2 + 1^2 = 10,$ and $\mathbf{u}\mathbf{u}^T = \begin{pmatrix} 3 \\ 1 \end{pmatrix} \begin{pmatrix} 3 & 1 \end{pmatrix} = \begin{pmatrix} 9 & 3 \\ 3 & 1 \end{pmatrix}.$
Therefore, the projection matrix for $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$ is $\frac{1}{10}\begin{pmatrix} 9 & 3 \\ 3 & 1 \end{pmatrix}.$
Similarly, for $\begin{pmatrix} 1 \\ 1 \end{pmatrix},$ I have $\|\mathbf{u}\|^2 = 1^2 + 1^2 = 2,$ and $\mathbf{u}\mathbf{u}^T = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \begin{pmatrix} 1 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}.$
Therefore, the projection matrix for $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$ is $\frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}.$
To find the matrix that takes $\mathbf{v}_0$ to $\mathbf{v}_2,$ I need to multiply the projection matrix for $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$ by the projection matrix for $\begin{pmatrix} 3 \\ 1 \end{pmatrix}.$
I use the standard formula for matrix multiplication, and I get

$\frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \frac{1}{10}\begin{pmatrix} 9 & 3 \\ 3 & 1 \end{pmatrix} = \frac{1}{20}\begin{pmatrix} 12 & 4 \\ 12 & 4 \end{pmatrix}.$
This is the matrix that takes $\mathbf{v}_0$ to $\mathbf{v}_2.$