I seek an example of a nonzero $\Bbb{R}$-bilinear map $f:V\times V\rightarrow W$ on a vector space $V$ (s.t: $\dim V<\infty$, $\dim W<\infty$) such that it is degenerate map, where $V$ and $W$ are given explicitly.
Recall that:
A degenerate bilinear form $f:V\times V\rightarrow W$ on a finite-dimensional vector space $V$ is a bilinear form such that it has a non-trivial kernel, i.e., there exist some non-zero $u$ in $V$ s.t $f(u,v)=0 $ for all $v\in V$.
Thank you in advance
Best Answer
recall if $f/V\times V \rightarrow W$ is a bilinear map and $W=K$ the field of scalar, then $f$ is sayd a bilinear fom.
if $dimV=1$ there are no degenerate bilinear map othere the zero map, so all non zero bilinear map are non degenerat.
if $dimV>1$ the example above in comment is on, anothere tack $f(e_1,e_j)=0$ and somme one $f(e_i,e_j)\not=0$ for $i>1$ and $j\geq 1$.