Let $W \subset \mathbb{R}^{4}$ a subspace generated by two vectors $$W := span \left\lbrace \begin{pmatrix} 1\\ 1\\ 0\\ 0 \end{pmatrix},\begin{pmatrix} 1\\ 1\\ 1\\ 2 \end{pmatrix} \right\rbrace. $$ Find $w \in W$ wich minimize $||w-v||$ where $v= \begin{pmatrix} 1\\ 2\\ 3\\ 4 \end{pmatrix}$. ($||\cdot ||$ is the usual norm in $\mathbb{R}^{4}$.
I found the projection matrix, and is
$$P= \begin{pmatrix} \frac{1}{2} & \frac{1}{2} & 0 & 0\\ \frac{1}{2} & \frac{1}{2} & 0 & 0\\ 0 & 0 & \frac{1}{5} & \frac{2}{5}\\ 0 & 0 & \frac{2}{5} & \frac{4}{5} \end{pmatrix} $$ so, $w= Pv=\begin{pmatrix} \frac{3}{2} \\ \frac{3}{2}\\ \frac{11}{5}\\ \frac{22}{5} \end{pmatrix}$ Am I right?, I don't know if I solved the problem.
Yes, your solution is correct and complete.