In $\mathbb{R}^{4}$ si considerino $u_{1}=\left(\begin{array}{r}1 \\ 2 \\ -3 \\ -2\end{array}\right), u_{2}=\left(\begin{array}{l}0 \\ 1 \\ 1 \\ 1\end{array}\right), w_{1}=\left(\begin{array}{r}3 \\ 7 \\ -8 \\ -5\end{array}\right), w_{2}=\left(\begin{array}{l}1 \\ 5 \\ 0 \\ 1\end{array}\right)$ e i sottospazi $U=\left\langle u_{1}, u_{2}\right\rangle$ e $W=\left\langle w_{1}, w_{2}\right\rangle$.
(a) Si determini $\operatorname{dim}(U+W)$.
(b) Si determini $\operatorname{dim}(U \cap W)$.
(Suggerimento: Si calcoli il rango della matrice avente le colonne $\left.u_{1}, u_{2}, w_{1}, w_{2} .\right)$
