[[Linear algebra MOC]]
# Complement subspace

Let $V$ be a [[vector space]] over $\mathbb{K}$ and $U \leq V$ be a subspace.
A **complement** $U^c \leq V$ is a subspace such that the [[Direct sum of vector spaces#internal direct sum|internal direct sum]] $U \oplus U^c = V$.

## Properties

1. Every $U \leq V$ has a (in general not unique) complement $U^c \leq V$.[^1] ^Existence

> [!check]- Proof of 1.
> The existence of the compliment follows from [[Every vector space has a basis]]:
> Let $\mathcal{A}$ be a basis of $U$.
> Then there exists a basis $\mathcal A$ of $V$ such that $\mathcal{A} \sube \mathcal{B}$.
> Then $U^c = \Span(\mathcal{B}\setminus \mathcal{A})$ is a complement of $U$. 
> <span class="QED"/>

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[^1]: This is downstream from [[Axiom of Choice|AC]].