# Anisotropic Vector Gives Composition of Bilinear Space

## Theorem

Let $\mathbb K$ be a field.

Let $\struct {V, f}$ be a bilinear space over $\mathbb K$.

Let $v \in V$ be anisotropic.

Let $\sequence v$ be its span.

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

Then $\struct {V, f}$ is the internal orthogonal sum of $\sequence v$ and $v^\perp$:

$V = \sequence v \oplus v^\perp$