Challenge Problems for SSEA 51, Homework 1
The first three problems are proofs, and you you should do your best to be rigorous.
(1) Let u, v, and w be linearly independent vectors in R3 . For which values of the scalar
a is the set {au + v − 4w, u + v, u + v + 2w} linearly independent? Prove your result.
(2) Prove that if {v1 , v2 , . . . , vk } is a set of k vectors in R2 , then span(v1 , v2 , . . . , vk ) is
either only the origin, a line through the origin, or all of R2 .
(3) Prove that there cannot be four linearly independent vectors in R3 .
(4) Explain how to do the magic trick with a die demonstrated in class. How does the
magician know whether the volunteer performed the last 90◦ rotation or not?
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