Proof
The
condition
implies
where the
matrix
consists of the columns
Hence,
Equivalently,
Therefore,
We next show
that
Indeed, we already established that
is a subspace, hence,
.
Therefore, if
then
.
But
and we proved already
that

Therefore,
According to the proposition (
Minimum
of a smooth
function
),
hence,
and the conclusion of the proposition follows.