Bilinear form/Nondegenerate/Definition

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ denote a ${\displaystyle {}K}$-vector space. A bilinear form

${\displaystyle V\times V\longrightarrow K,(v,w)\longmapsto \left\langle v,w\right\rangle ,}$

is called nondegenerate, if for every ${\displaystyle {}v\in V}$, ${\displaystyle {}v\neq 0}$, the induced mapping

${\displaystyle V\longrightarrow K,w\longmapsto \left\langle v,w\right\rangle ,}$

and, for every ${\displaystyle {}w\in V}$, ${\displaystyle {}w\neq 0}$, the induced mapping

${\displaystyle V\longrightarrow K,v\longmapsto \left\langle v,w\right\rangle ,}$

is not the zero mapping.