# Intersection Condition for Direct Sum of Subspaces

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## Theorem

Let $U$ and $W$ be subspaces of a vector space $V$.

Then $U + W$ is a direct sum if and only if $U \cap W = 0$.

## Proof

We must first prove that if $U + W$ is a direct sum, then:

- $U \cap W = 0$

Let $U + W$ be a direct sum.

Let $\mathbf v \in U \cap W$ be an arbitrary vector in $U \cap W$.

Then:

- $\mathbf 0 = \mathbf v + \paren {-\mathbf v}$

where:

- $\mathbf v \in U$
- $-\mathbf v \in W$

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## Also see