Proof
We show that both compositions are the identity. We start with a matrix
and consider the matrix
-
Two matrices are equal, when the entries coincide for every index pair
. We have

Now, let
be a linear mapping, we consider
-
Two linear mappings coincide, due to
fact,
when they have the same values on the basis
. We have
-

Due to the definition, the coefficient
is the
-th coordinate of
with respect to the basis
. Hence, this sum equals
.