We show first that the new family is a generating system. Because of
-
and
,
we can express the vector as
-
Let
be given. Then, we can write
To show the
linear independence,
we may assume
to simplify the notation. Let
-
be a representation of . Then
-
From the linear independence of the original family, we deduce
.
Because of
,
we get
.
Therefore,
,
and hence
for all .