## 403/503 – HW2

Let $V$ be a vector space. In lecture we verified that the following two statements about a set $A\subset V$ are equivalent:

• For any $v_1,\dots,v_n\in A$ and any scalars $r_1,\dots,r_n$ and $s_1,\dots,s_n$, if $\displaystyle \sum_{i=1}^n r_i v_i=\sum_{i=1}^n s_i v_i$, then $r_i=s_i$ for all $i$.
• For any $v_1,\dots,v_n\in A$ and any scalars $r_1,\dots,r_n$, if $\displaystyle \sum_{i=1}^n r_i v_i=0$, then $r_i=0$ for all $i$.

Recall that the set $A$ is independent iff no element of $A$ is in the span of the other elements, that is, for any $a\in A$, we have that $a\notin\mathrm{sp}(A\setminus\{a\})$.

1. Show that $A$ is independent iff the two (equivalent) statements above hold.

(Due January 22 at the beginning of lecture.)