How do you determine dimension Suppose $U$ and $V$ are $2$ dimensional subspaces of $\mathbb{R}^4$. How do you determine the dimension of $U \cap V$? I know that 
\begin{equation*}
\text{dim}(U + V) = \text{dim}(U)+ \text{dim}(V) - \text{dim}(U \cap V). 
\end{equation*}
So it seems that $\text{dim}(U \cap V) = 0$.

If you have B and B', basis of U and V respectively, you can check if the vectors of B are linearly independent with those of B'. If so, then $\dim(U\cap V) = 0$. If not, you may be able to write a new basis of $U+V$ eliminating the linearly dependent vectors. $\dim(U\cap V)$ is the amount of vectors you have eliminated from $U+V$'s basis.
Also, $\dim(U+V) = \dim(U) + \dim(V) - \dim(U\cap V)$.
