Anisotropic Vector Gives Composition of Bilinear Space

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Theorem

Let $\mathbb K$ be a field.

Let $\left({V, f}\right)$ be a bilinear space over $\mathbb K$.

Let $v \in V$ be anisotropic.

Let $\left\langle{v}\right\rangle$ be its span.

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


Then $\left({V, f}\right)$ is the internal orthogonal sum of $\left\langle{v}\right\rangle$ and $v^\perp$:

$V = \left\langle{v}\right\rangle \oplus v^\perp$


Proof