Let $\varphi: E \times F \to G$ be a bilinear mapping. Show that the following property is equivalent to $\otimes_2$: Whenever the vectors $x_{\alpha}\in E$ and $y_{\beta} \in F$ are linearly independent then so are the vectors $\varphi(x_{\alpha},y_{\beta})$
Property $\otimes_2$ is as follows:
I don't understand how this equivalence is maintained. I know that by the linear independence of the $x_{\alpha} $ in $E$ and the $y_{\beta}$ in $F$, all non-zero vector $g \in G$ can be written in the form $g = \sum_{i = 1}^{r} {x_{\alpha_{i}} \otimes y_{\beta_{i}}} $. But, I'm not totally sure of this equivalence. I would like to know if there is a proof that justifies this statement.