If $V_{1}$ and $V_{2}$ are vector spaces, what is meant by $V_{1} \oplus V_{2}$ ? If $V_{1}$ and $V_{2}$ are subspaces of a vector space $V$, what is meant by $V_{1}+V_{2}$ ?

Stating clearly any theorems you use, show that if $V_{1}$ and $V_{2}$ are subspaces of a finite dimensional vector space $V$, then

$$\operatorname{dim} V_{1}+\operatorname{dim} V_{2}=\operatorname{dim}\left(V_{1} \cap V_{2}\right)+\operatorname{dim}\left(V_{1}+V_{2}\right)$$

Let $V_{1}, V_{2} \subset \mathbb{R}^{4}$ be subspaces with bases

$$\begin{gathered} V_{1}=\langle(3,2,4,-1),(1,2,1,-2),(-2,3,3,2)\rangle \\ V_{2}=\langle(1,4,2,4),(-1,1,-1,-1),(3,1,2,0)\rangle . \end{gathered}$$

Find a basis $\left\langle\mathbf{v}_{1}, \mathbf{v}_{2}\right\rangle$ for $V_{1} \cap V_{2}$ such that the first component of $\mathbf{v}_{1}$ and the second component of $\mathbf{v}_{2}$ are both 0 .