Let $V$ be a [[finite dimensional vector space]] over the [[field]] $K$. Let $W$ be a [[subspace]]. Then there exists a subspace $U$ such that $V$ is the direct sum of $W$ and $U$. #### Proof Let $V$ have a basis $\set{v_1,\cdots,v_n}$ Let $W$ have a basis $\set{v_1,\cdots,v_r}$ The [[completing a basis|existence of a completed basis]] allows the basis for $U$ to take the set-subtraction of the basis $\set{v_1,\cdots,v_n}\setminus\set{v_1,\cdots,v_r}=\set{v_{r+1},\cdots,v_n}$