If $U$ and $W$ are finite-dimensional subspaces of a vector space $V$, prove that

$$\operatorname{dim}(U+W)=\operatorname{dim}(U)+\operatorname{dim}(W)-\operatorname{dim}(U \cap W)$$

Let

$$\begin{aligned} U &=\left\{\mathbf{x} \in \mathbb{R}^{4} \mid x_{1}=7 x_{3}+8 x_{4}, x_{2}+5 x_{3}+6 x_{4}=0\right\} \\ W &=\left\{\mathbf{x} \in \mathbb{R}^{4} \mid x_{1}+2 x_{2}+3 x_{3}=0, x_{4}=0\right\} . \end{aligned}$$

Show that $U+W$ is 3 -dimensional and find a linear map $\ell: \mathbb{R}^{4} \rightarrow \mathbb{R}$ such that

$$U+W=\left\{\mathbf{x} \in \mathbb{R}^{4} \mid \ell(\mathbf{x})=0\right\}$$