Ps3 - Problem Set 3 Math 115A/5 Fall 2011 Due Friday October 21 Problem 1 Consider the function T P(F F(F F which takes a polynomial f(x = an xn a0

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Problem Set 3 Math 115A/5 — Fall 2011 Due: Friday, October 21 Problem 1 Consider the function T : P ( F ) → F ( F,F ) which takes a polynomial f ( x ) = a n x n + ··· + a 0 with coefficients in F to the corresponding function f : F F obtained by evaluation of the polynomial. (a) Show that T is a linear transformation. (b) Show that, if F is infinite, T is injective. (c) For F = F 2 , the field of 2 elements, find a polynomial in the null space of T . Problem 2 (1.6.20) Let V be a vector space having dimension n , and let S be a subset of V that generates V . (a) Prove that there is a subset of S that is a basis for V . (b) Prove that S contains at least n vectors. Problem 3 (1.6.33) (a) Let W 1 and W 2 be subspaces of a vector space V such that V = W 1 W 2 . If β 1 and β 2 are bases for W 1 and W 2 , respectively, show that β 1 β 2 = and β 1 β 2 is a basis for V . (b)
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This note was uploaded on 04/09/2012 for the course MATH 172a taught by Professor Kong,l during the Fall '08 term at UCLA.

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Ps3 - Problem Set 3 Math 115A/5 Fall 2011 Due Friday October 21 Problem 1 Consider the function T P(F F(F F which takes a polynomial f(x = an xn a0

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