Matrix multiplication/Composition/Fact/Proof/Exercise

Let ${\displaystyle {}B}$ be an ${\displaystyle {}n\times p}$-Matrix, and ${\displaystyle {}A}$ be an ${\displaystyle {}m\times n}$-matrix. Let

${\displaystyle K^{p}{\stackrel {B}{\longrightarrow }}K^{n}{\stackrel {A}{\longrightarrow }}K^{m}}$

denote the corresponding linear mappings. Show that the matrix product ${\displaystyle {}A\circ B}$ describes the composition of the linear mappings.