Dimension of Radical of Bilinear Form

Theorem

Let $\mathbb K$ be a field.

Let $V$ be a vector space over $\mathbb K$ of finite dimension $n > 0$.

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

Let $\map \Rad V$ denote the radical of $V$.

Let $\map {\operatorname {rk} } f$ be the rank of $f$.

Then:

$\map \dim {\map \Rad V} = n - \map {\operatorname {rk} } f$

where $\dim$ denotes dimension.

Proof

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