Let be a vector space. In lecture we verified that the following two statements about a set are equivalent:
- For any and any scalars and , if , then for all .
- For any and any scalars , if , then for all .
Recall that the set is independent iff no element of is in the span of the other elements, that is, for any , we have that .
- Show that is independent iff the two (equivalent) statements above hold.
(Due January 22 at the beginning of lecture.)