Dimension of Orthogonal Complement With Respect to Bilinear Form

Theorem

Let $\mathbb K$ be a field.

Let $V$ be a vector space over $\mathbb K$ of finite dimension.

Let $f$ be a nondegenerate bilinear form on $V$.

Let $U\subset V$ be a subspace.

Let $U^\perp$ be its orthogonal complement.

Then:

$\map \dim U + \map \dim U^\perp = \map \dim V$

Proof

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