= Zero-composition subspaces of a linear map
{title2=$M^l(\alpha)=\{\beta:\beta\alpha=0\},\quad M^r(\alpha)=\{\beta:\alpha\beta=0\}$}
For a <linear map> $\alpha:U\to V$ between <finite-dimensional vector spaces>, take $\beta\in\operatorname{Hom}(V,U)$. The <vector subspace> $M^l(\alpha)=\{\beta:\beta\alpha=0\}$ consists of the <linear maps> factoring uniquely through $V/\operatorname{im}\alpha$, whereas $M^r(\alpha)=\{\beta:\alpha\beta=0\}$ consists of those whose <image of a linear map> lies in $\ker\alpha$. Thus
$$
M^l(\alpha)\cong\operatorname{Hom}(V/\operatorname{im}\alpha,U),\qquad M^r(\alpha)\cong\operatorname{Hom}(V,\ker\alpha).
$$
Writing $u=\dim U$, $v=\dim V$ and $r=\operatorname{rank}\alpha$, 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.
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