TITLE: Example of a bilinear map whose image is not a subspace
QUESTION [5 upvotes]: I am looking for an example of a bilinear map $\tau:V \times V \to W$ whose image $im(\tau)=(\tau(u,v):u,v \in V)$ is not a subspace of $W$.
I considered the tensor map $\tau:U \times V \to U \otimes V$, since its images consists of all decomposable tensors. I have the idea that if $u \otimes v, u'\otimes v'$ are decomposable tensors, then $u \otimes v + u'\otimes v'$ is not necessarily a decomposable vector, but I am not sure why.

REPLY [11 votes]: The image of $* : K[x] \times  K[y] \to K[x,y]$ contains $x$ and $y$, but not $x+y$.