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.)

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The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]

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