403/503 – HW2

January 20, 2015

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.)