Matrices/Multiplication condition/Interpretation/Exercise

Let ${}K$ be a field.

a) Let ${}L$ be an ${}\ell \times m$-matrix, and let ${}N$ be an ${}n\times p$-matrix. Show that

${\left\{M\mid M\in \operatorname {Mat} _{m\times n}(K),\,L\circ M\circ N=0\right\}}$

is a linear subspace of ${}\operatorname {Mat} _{m\times n}(K)$.

b) Let ${}V$ be an ${}n$-dimensional and ${}W$ be an ${}m$-dimensional ${}K$-vector space, and let ${}U\subseteq V$ and ${}T\subseteq W$ denote linear subspaces. Describe the linear subspace

${\left\{\varphi \in \operatorname {Hom} _{K}{\left(V,W\right)}\mid \varphi (U)\subseteq T\right\}}$

using suitable bases and part a).

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