Zero-composition subspaces of a linear map ($M^l(\alpha )=\{\beta :\beta \alpha =0\},M^r(\alpha )=\{\beta :\alpha \beta =0\}$)

For a linear map $\alpha :U\to V$ between finite-dimensional vector spaces, take $\beta \in \mathrm{Hom}(V,U)$. The vector subspace $M^l(\alpha )=\{\beta :\beta \alpha =0\}$ consists of the linear maps factoring uniquely through $V/\mathrm{im}\alpha$, whereas $M^r(\alpha )=\{\beta :\alpha \beta =0\}$ consists of those whose image of a linear map lies in $\ker \alpha$. ThusWriting $u=\dim U$, $v=\dim V$ and $r=\mathrm{rank}\alpha$, their dimensions are $(v-r)u$ and $(u-r)v$. Passing to the dual map reverses composition and interchanges the corresponding left and right conditions. The finite-dimensional hypothesis makes these dimension differences ordinary integer subtraction.