235HW8 - Math 235 Linear Algebra HW 8 Problem 3.6.1 Problem 3.6.1 Let n be a positive integer and F a eld Let W be the set of all vectors(x1 xn in

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Math 235: Linear Algebra HW 8 Problem 3.6.1 Problem 3.6.1 Let n be a positive integer and F a field. Let W be the set of all vectors ( x 1 ,...,x n ) in F n such that x 1 + x 2 + ··· + x n = 0. (a) Prove that W 0 consists of all linear functaionals f of the form f ( x 1 ,...,x n ) = c n X j =1 x i Let S = ( f ( x 1 ,...,x n ) = c n X i =1 x i ) . Clearly for every w W and f S we have that f ( w ) = c n X i =1 x i = c · 0 = 0 so S W 0 . Now we must show that S is all of W 0 , we’ll show this using dimensions. dim ( W ) = n - 1 (we can pick x 1 ,...,x n - 1 then x n = - n - 1 X i =1 x i ). We know that dim W + dim W 0 = dim F n = n n - 1 + dim W 0 = n dim W 0 = 1. Clearly dim S = 1 so we get that S = W 0 . (a) (b) Show that the dual space W * of W can be ’naturally’ identified with the linear functionals f ( x 1 ,...,x n ) = c 1 x 1 + ··· + c n x n on F n which satisfy c 1 + ··· + c n = 0. This is actually quite easy. Let w = ( w 1 ,...,w n ) W then w 1 + w 2 + ··· + w n = 0, so we get the linear functional f = w 1 x 1 + w 2 x 2 + ··· + w n x n . Similarly taking a linear funcional f = c 1 x 1 + c 2 x 2 + ··· + c n x n where c 1 + ··· + c n = 0 we get the vector w = ( c 1 ,...,c n ) which is in W since c 1 + ··· + c n = 0. (b) Page 1 of 5
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Math 235: Linear Algebra HW 8 Problem 3.6.2 Problem 3.6.2 Use Theorem 20 to prove the following. If W is a subspace of a finite-dimensional vector space V and if { g 1 ,...,g r } is any basis of W 0 , then W = r \ i =1 N g i For every f W 0 we have that f = r X n =1 a i g i so by theorem 20
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This note was uploaded on 11/14/2010 for the course PHYCS 498 taught by Professor Aa during the Spring '10 term at University of Illinois, Urbana Champaign.

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235HW8 - Math 235 Linear Algebra HW 8 Problem 3.6.1 Problem 3.6.1 Let n be a positive integer and F a eld Let W be the set of all vectors(x1 xn in

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