Recall that a vector space is said to be of dimension
,
, iff there is an independent subset of
of size
, and no independent subset has size
.
A basis of a vector space is any independent set whose span is
.
Suppose that is a vector space of dimension
. In lecture we showed that a subset of
of size
is independent iff it spans
. Show the following:
has a basis, and all bases have size
.
- If
is independent, then there is a basis
of
with
.
(Due January 27 at the beginning of lecture.)