I've been trying to delve a little further into linear algebra, but I'm not following something I think is supposed to be obvious.
Suppose $M_{m,n}(\mathbb{C})$ is the set of rectangular $m\times n$ matrices over $\mathbb{C}$, and let $S$ be the set of rank $1$ matrices. Furthermore, let $K\subset\mathbb{C}[S_{11},\dots,S_{mn}]$ be the ideal associated to $S$. Then the homomorphism $\mathbb{C}[S_{11},\dots,S_{mn}]\to\mathbb{C}[X_1,\dots,X_m,Y_1,\dots,Y_n]$ such that $S_{ij}\mapsto X_iY_j$ has kernel $K$.
I don't follow the last claim. I'm used to the associated ideal of $S$ to be the polynomials in $\mathbb{C}[S_{11},\dots,S_{mn}]$ to be the ideal of polynomials which vanish on all points of $S$ for an algebraic set of zeroes, but that doesn't quite make sense with a set of rank $1$ matrices. Would someone be nice enough to explain why the kernel above is what it is? Thank you.