CHAPTER 2 Finite-Dimensional Vector Spaces
- If some vectors are removed from a linearly independent list, the remaining
list is also linearly independent, as you should verify.
Proof Suppose is linearly independent, then the only choice of
that makes
is
.
Since the order of vectors does not affect linear independence, we put all the vectors that need to be removed on the right.
Now assume the vectors are removed, and the remaining
is linearly dependent.
Hence there exist , not all
, such that
.
Let all are
, hence
, at the same time
, not all
. We get a contradiction.
Thus the remaining lise is also linearly independent.
