Maps between projective spaces induced by singular matrices

Question

In this question, $V(a,k)$ denotes the $a$-dimensional vector space over the field $k$. Now consider $V(b,k)$ and $V(c,k)$, and let $M$ be a singular $(c \times b)$-matrix over $k$. Let $\ell_M: V(b,k) \mapsto V(c,k)$ be the linear transformation induced by $M$. What kind of map does $\ell_M$ induce between the projective spaces $\mathbb{P}^{b - 1}(k)$ and $\mathbb{P}^{c - 1}(k)$ (seen as projective varieties)? I guess it is a rational map, but what are the most interesting, or useful/important, examples of maps of this type ?