Example of a degenerate bilinear map?

Question

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

Answer 1

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 $j\geq 1$.

Answer 2

A bilinear form $f:V\times V \to \mathbb{K}$ takes values in the underlying field of scalars $\mathbb{K}$ of $V$, and is entirely determined by its action on pairs of basis elements of $V$. If $V$ is finite-dimensional, $\dim V = n$, then each choice of basis $(e_i)_{i=1}^n$ for $V$ induces an isomorphism between the space of bilinear forms and the space of $n\times n$ matrices with values in $\mathbb{K}$: the isomorphism maps the form $f$ to the matrix $(f(e_i,e_j))$, and its inverse maps the matrix $A$ to the form $((u,v) \mapsto u^t A v)$.

This yields as many examples of bilinear forms as there are $n \times n$ matrices. Moreover, under this isomorphism, degenerate forms correspond to singular matrices. So for $n = 2$, for instance, a form $f$ is degenerate iff $\det (f(e_i,e_j)) = f(e_1,e_1)f(e_2,e_2) - f(e_1,e_2)f(e_2,e_1) = 0$.