---- > [!definition] Definition. ([[nondegenerate bilinear form]]) > Let $V$ be a [[vector space]] over a [[field|field]] $\mathbb{F}$ and let $\langle -,- \rangle$ be a (say, [[symmetric multilinear map|symmetric]]) [[bilinear map|bilinear form]] on $V$. We call the form **nondegenerate** if its [[orthogonal complement|kernel]] is zero, i.e., if for all $v \in V$, the map $\langle v, - \rangle: V \to \mathbb{F}$ is the zero map if and only if $v=0$. > > > > Note that if $V$ is finite-dimensional, then the form is nondegenerate iff $v \mapsto \langle v,- \rangle$ is an [[isomorphism]] iff its [[matrix of a bilinear form|matrix]] is [[inverse matrix|invertible]]. See also [[musical isomorphism induced by a nondegenerate bilinear form]]. > > ^definition > [!generalization] > $\langle -,- \rangle$ is nondegenerate precisely when $(V, V, \langle -,- \rangle)$ is a [[orthogonal complement|dual pair]]. ^generalization [^1]: Since $V$ finite-dimensional, $\dim V=\dim V^{*}$ and checking this claim amounts to checking that the kernel of $v \mapsto \langle v,- \rangle$ is trivial. But this ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```