403/503 – HW3

January 23, 2015

Recall that a vector space V is said to be of dimension n, \dim V=n, iff there is an independent subset of V of size n, and no independent subset has size n+1.

A basis of a vector space V is any independent set whose span is V.

Suppose that V is a vector space of dimension n. In lecture we showed that a subset of V of size n is independent iff it spans V. Show the following:

  1. V has a basis, and all bases have size n.
  2. If A\subseteq V is independent, then there is a basis B of V with A\subseteq B.

(Due January 27 at the beginning of lecture.)