nov16 - U is free on v , v , we may dene f , f : U F by f (...

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Math 5125 Wednesday, November 16 November 16, Ungraded Homework Exercise 10.4.12 on page 376 Let V be a vector space over the field F and let v , v 0 be nonzero elements of V . Prove that v v 0 = v 0 v in V F V if and only if v = av 0 for some a F . Suppose we do not have v = av 0 for some a F . Then v , v 0 are linearly independent and therefore span a subspace U of V which has dimension two. Write V = U W where W is a subspace of V (every subspace of a vector space has a direct complement; for infinite dimensional vector spaces this requires the axiom of choice). Since
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Unformatted text preview: U is free on v , v , we may dene f , f : U F by f ( v ) = f ( v ) = 1 and f ( v ) = f ( v ) = 0. We may extend f , f to the whole of V by dening f ( w ) = f ( w ) = 0 for all w W . Then f f ( v v ) = 1 and f ( v v ) = 0. Applying f f to v v = v v , we deduce that 1 = 0 which is a contradiction and the result is proven....
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This note was uploaded on 01/02/2012 for the course MATH 5125 taught by Professor Palinnell during the Fall '07 term at Virginia Tech.

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