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Solution03_121

# Solution03_121 - Problem Set 3 Solutions J Scholtz Question...

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Problem Set 3 - Solutions J. Scholtz October 24, 2006 Question 1: 1.3/31 a) Cosets and Subspaces We want to show that v + W is a subspace if and only if v W . ( ) Suppose that v + W is a subspace. v + W must contain 0. Then there exists u W such that v + u = 0, hence W contains - v , and sincd it is a subspace itself then W contains also v . ( ) If v W , then the set of form { v + w, w W } = W , since that is closed under addition. Therefore v + W = W which we know is a subspace. b) Equivalent representations Suppose that v 1 + W = v 2 + W , then this is equivalent to the statement that for every w 1 W , there exists w 2 W such that v 1 + w 1 = v 2 + w 2 , which is equivalent with the statement v 1 - v 2 = w 2 - w 1 , but this is equivalent with the statement that v 1 - v 2 W since w 2 - w 1 W since W is a suspace. Since all these statements were equivalent then the if and only condition is satisfied. c) Are these operations well defined? If v 1 + W = v 1 + W and v 2 + W = v 2 + W then v 1 - v 1 W and v 2 - v 2 W . Therefore ( v 1 + v 2 ) - ( v 1 + v 2 ) W , hence ( v 1 + v 2 )+ W = ( v 1 + v 2 )+ W , which in turn means that ( v 1 + W ) + ( v 2 + W ) = ( v 1 + W ) + ( v 2 + W ). For scalar multiplication: if v 1 + W = v 1 + W then v 1 - v 1 W , hence a ( v 1 - v 1 ) W therefore av 1 - av 1 W which in turn implies that av 1 + W = av 1 + W . But this is equivalent to the statement: a ( v 1 + W ) = a ( v 1 + W ). d) They form a vector space! We need to check all the axioms: VS1(Commutativity): ( v + W ) + ( u + W ) = ( v + u ) + W = ( u + v ) + W = ( u + W ) + ( v + W ), where we used the fact that V is a vectorspace. VS2(Associativity): Use the same trick knowing that V is a vectorspace. VS3(Existence of 0): The coset 0 + W does the job since (0 + W ) + ( a + W ) = ( a + 0) + W = a + W . 1

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VS4(Inverses): Use the inverses from V : ( - v + W ) + ( v + W ) = ( v - v ) + W = 0 + W . VS5(Identity): Take the identity from V : 1( v + W ) = 1 v + W = v + W . The last three are inherited due to the nature of definition of addition and scalar multiplication in V/W . Question 2: 1/6/33 a) Let V = W 1 W 2 , that means that W 1 W 2 = 0. Now suppose that v β 1 β 2 , this would mean that v W 2 and v W 1 , which is a contradiction.
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