If $V$ is a [[finite dimensional vector space]] over [[field]] $K$, and is the [[direct sum of subspaces]] of subspaces $U, W$ then $ \operatorname{dim} V=\operatorname{dim} U+\operatorname{dim} W $ #### Proof Let $\left\{u_{1}, \ldots, u_{r}\right\}$ be a basis of $U$. Let $\left\{w_{1}, \ldots, w_{s}\right\}$ be a basis of $W$. The [[uniqueness of coordinate in a basis]] suggests both $x_{1} u_{1}+\cdots+x_{r} u_{r}$ and$y_{1} w_{1}+\cdots+y_{s} w_{s}$ sum to $x_{1} u_{1}+\cdots+x_{r} u_{r}+y_{1} w_{1}+\cdots+y_{s} w_{s}$ With all terms being independent, showing $\set{u_{1}, \ldots, u_{r}, w_{1,} \ldots, w_{s}}$ is a basis for $V=U\oplus W$ $ \operatorname{dim} V=\operatorname{dim} U+\operatorname{dim} W \quad \begin{cases}\operatorname{dim}U=r\\ \operatorname{dim}W=s\\ \operatorname{dim}V=r+s\end{cases} $