Symmetric bilinear form/Nondegenerate/Gradient/Definition

Let ${\displaystyle {}V}$ be a finite-dimensional ${\displaystyle {}K}$-vector space, and let ${\displaystyle {}\left\langle -,-\right\rangle }$ denote a nondegenerate symmetric bilinear form on ${\displaystyle {}V}$. For a given linear form

${\displaystyle L\colon V\longrightarrow K,}$

the uniquely determined vector ${\displaystyle {}z\in V}$ fulfilling

${\displaystyle {}L(v)=\left\langle z,v\right\rangle =\left\langle v,z\right\rangle \,}$

for all ${\displaystyle {}v\in V}$, is called the gradient of ${\displaystyle {}L}$ with respect to the bilinear form.