Proof
Set
.
Let
denote the
kernel
of the mapping and let
denote its
dimension
().
Let
-
be a
basis
of . Due to
fact,
there exist vectors
-
such that
-
is a basis of .
We claim that
-
is a basis of the image. Let
be an element of the image . Then there exists a vector
such that
.
We can write with the basis as
-
Then we have
which means that is a
linear combination
in terms of the .
In order to prove that the family
, ,
is
linearly independent,
let a representation of zero be given,
-
Then
-
Therefore, belongs to the kernel of the mapping. Hence, we can write
-
Since this is altogether a basis of , we can infer that all coefficients are , in particular,
.