## 403/503 – HW3

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.)