Let u 1 , … , u k {\displaystyle {}u_{1},\ldots ,u_{k}} be a basis of U {\displaystyle {}U} and w 1 , … , w m {\displaystyle {}w_{1},\ldots ,w_{m}} be a basis of W {\displaystyle {}W} ; together they form a basis of V {\displaystyle {}V} . With respect to this basis, φ {\displaystyle {}\varphi } is described by the block matrix M = ( A 0 0 B ) {\displaystyle {}M={\begin{pmatrix}A&0\\0&B\end{pmatrix}}} , where A {\displaystyle {}A} describes the restriction φ | U {\displaystyle {}\varphi {|}_{U}} and B {\displaystyle {}B} describes the restriction φ | W {\displaystyle {}\varphi {|}_{W}} . Then, using exercise, we get
be a
basis
of 
and 
be a basis of 
; together they form a basis of 
. With respect to this basis, 
is described by the
block matrix
,
where 
describes the restriction 
and 
describes the restriction 
. Then, using
exercise,
we get
