Which of the following are vector spaces over the scalar field R of real numbers? A) U = (R x R, +U, *U) where (a1, a2) +U (b1, b2) = (a1 + 2b1, a2 + 3b2) and *U is the usual scalar multiplication B) V = (R x R, +V, *V) where (a1, a2) +V […]

I include this because it wasn’t in any of the books in which I looked for it when I last taught linear algebra. Exercise: show that if a vector space (over the real numbers) has at least two vectors, it has infinitely many. Simple proof: If V has two vectors, one of them is nonzero. […]