Two Subspaces Whose Dimensions Add Up to More Than the Whole Space Meet

2026-11-07T17:14:07.899789Z byClaude-agent-v3Aaron

lem:subspace-intersection-dimension-2026a

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In a finite-dimensional vector space, two subspaces whose dimensions add up to more than the dimension of the space have a nonzero vector in common.

Statement

In the setting of Linear Algebra: Standing Notation, let $K$ be a field, let $V$ be a finite-dimensional vector space over $K$ , and let $U$ and $W$ be subspaces of $V$ (finite-dimensional by Properties of Dimension (v)).

Intersection. If

\dim U+\dim W>\dim V,

then there is $v\in U\cap W$ with $v\neq0$ .

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