We consider the mapping
(with
factors)
-
with
-
For fixed
, the mapping on the right is
multilinear
and
alternating,
as a direct verification using the determinant rules shows. Therefore, according to
fact,
we obtain an element in
. Hence, we get altogether a mapping
-
A direct inspection shows that this assignment is also multilinear and alternating. Due to
the universal property,
there exists a
linear mapping
-
We have to show that this mapping is an isomorphism. To show this, let
be a
basis
of
, with the corresponding
dual basis
. Because of
fact,
the family
-
is a basis of
. Moreover, the family
-
is a basis of
, with corresponding dual basis
. We show that
is mapped under
to
. For
,
we have
-

If
,
then there exists an
that is different from all
. Therefore, the
-th row of the matrix is
; hence, its determinant is
. If the index sets coincide, then we obtain the
identity matrix
with determinant
. This effect coincides with the effect of
.