= Solution
Take $V=U=\mathbb R^3$ and identify $V\otimes_{\mathbb R}U$ with the space of $3\times3$ real matrices, where <pure tensor>[pure tensors] correspond to matrices of rank at most one. Let
$$
W=(V\otimes U)/\mathbb R I_3
$$
and let $f$ be the quotient map. It is not injective because its kernel is the nonzero line $\mathbb RI_3$.
If $f(v_1\otimes u_1)=f(v_2\otimes u_2)$, then
$$
v_1\otimes u_1-v_2\otimes u_2=\lambda I_3
$$
for some $\lambda\in\mathbb R$. The left side has matrix rank at most two. If $\lambda\ne0$, the right side has rank three, which is impossible. Thus $\lambda=0$ and the two pure tensors were equal. Hence $f$ is injective on the set of pure tensors while failing to be injective linearly.
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