bilinear transformation $\phi U\times V\to W$ such that $Im(\phi)=\{\phi(u,v): u\in U, v\in V\}$ is not a subspace of $W$ Find a bilinear transformation $\phi U\times V\to W$ such that $Im(\phi)=\{\phi(u,v): u\in U, v\in V\}$ is not a subspace of $W$
I truly don't have an idea otherwise to brute force lots of tries and find one that fits. Is there a technique of some sort that can help?

As far as I know there is no technique, but you might want to consider the case $U=V=\Bbb{R}^2$ and the map $\phi$ that sends a pair to the four coordinate products. That is to say
$$\phi:\ \Bbb{R}^2\times\Bbb{R}^2\ \longrightarrow\ \Bbb{R}^4: ((x_1,y_1),(x_2,y_2))\ \longmapsto\ (x_1x_2,x_1y_2,y_1x_2,y_1y_2).$$
