> [!NOTE] Definition (Complement of a subspace) > Let $V$ be a [[Vector Space|vector space]] and $U\subset V$ a [[Vector Subspace|subspace]] of $V.$ Then a subspace $U'\subset V$ is called a **complement** to $U$ iff $V=U+U',$ their [[Sum of Vector Subspaces|sum]] which is the span of their union, and $U \cap U'=\{ 0_{V} \}.$ # Properties > [!NOTE] Corollary (Dimension of complement) > If $U'$ is a complement to $U\subset V$, then $\dim V = \dim U + \dim U'$ ^6489d8 >Proof. Follows from [[Dimension of Sum of Finite Dimensional Vector Subspaces (Dimension Formula)|dimension formula]], $U + U'=V$ and $U \cap U' =\{ 0_{V} \}$. # Applications The third proof of the [[Rank-Nullity Formula|Rank–Nullity Theorem]] uses complements.