Tensor product and span

Question

Let $\varphi(u,v) = u \otimes v$, prove that $\varphi(U\times V)$ spans $U \otimes V$, where $U, V$ are vector spaces over a field $K$.

I defined a function $i: U \times V \rightarrow U \otimes V$ s.t $i(u, v) = u \otimes v$, so by the universal property there is a unique function $h: U \otimes V \rightarrow U \otimes V$ s.t $h(u \otimes v) = u \otimes v = Id(u \otimes v)$ and $i = h \circ \varphi$.

Now, $h = Id$ is surjective, so $i$ must be surjective, and that completes the proof.

Is it right?

Answer 1

Hint: It suffices to show the following. For linear maps $f_1,f_2:U \otimes V \to U \otimes V$, if $f_1|_{\varphi(U \times V)} = f_2|_{\varphi(U \times V)}$, then $f_1 = f_2$. Note that $f_i|_{\varphi(U \times V)}$ define bilinear maps on $U \times V$.