#math/linear-algebra The dimension of a finite-dimensional vector space, $V$, is the length of _any_ basis of $V$. Additionally, if $U$ is a subspace of $V$, then $\text{dim } U \leq{} \text{dim} V$. If $U_{1}$ and $U_{2}$ are subspaces of a finite-dimensional vector space $V$ then $\text{dim }V=\text{dim }U_{1}+ \text{dim }U_{2}-\text{dim }(U_{1}\cap U_{2})$.