Let $T_2=(\mathbf{M}(\mathbb{F});+;\cdot)$ be the ring of $n\times m$ matrices over the field $\mathbb{F}$ in the usual sense. We denote elements in $T_2$ as $M(n;m)$. If we consider $T_2$ as rank 2 tensors, and let the set $$X=\{φ|φ\in B:T_2\times T_2\rightarrow\mathbb{F}\}$$ be the set of all bilinear forms in $T_2$, then, clearly the set:
$$S=\{x\cdot y|x=M(1;k); y=M(k;1);\\ k=1,2,…,n\}$$ is a trivial subset of X, and acts like a contraction $V\otimes V^*\in\mathbb{F}$ in $T_2$. Is the inclusion $S\subseteq X$ a proper inclusion? Are there any non-trivial bilinear forms in $T_2$ wich are not in $S$?