Non-degenerate bilinear form in the context of Lorentzian manifold

In mathematics, specifically linear algebra, a degenerate bilinear form ${\displaystyle f(x,y)}$ on a vector space ${\displaystyle V}$ is a bilinear form such that the map from ${\displaystyle V}$ to ${\displaystyle V^{*}}$ (the dual space of ${\displaystyle V}$) given by ${\displaystyle v\mapsto (x\mapsto f(x,v))}$ has a non-trivial kernel, i.e. there exist some non-zero ${\displaystyle x}$ in ${\displaystyle V}$ such that ${\displaystyle f(x,y)=0}$ for all ${\displaystyle y\in V}$.

An equivalent definition when ${\displaystyle V}$ is finite-dimensional is that the previous map is not an isomorphism.