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.)