108AQuiz2 - ,...,T ( w n )). 2. Prove that the list ( T ( w...

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Name: Mathematics 108A: Quiz 2 Friday, May 7, 2010 Professor J. Douglas Moore One of the Main Theorems from Chapter 2 of the text is: Theorem. If V is a finite dimensional vector space and T : V W is a linear map into a vector space W , then dim V = dim N ( T ) + dim R ( T ) . Idea of proof: We start by choosing a basis ( u 1 ,..., u m ) for N ( T ). The Exten- sion Theorem states that we can extend this to a basis ( u 1 ,..., u m , w 1 ,..., w n ) of V . If we can show that ( T ( w 1 ) ,...,T ( w n )) is a basis for R ( T ), then dim R ( T ) = n . It will then follow that dim V = m + n = dim N ( T ) + dim R ( T ) , and the theorem will be proven. Thus we need only carry out the following two steps: 1. Prove that the list ( T ( w 1 ) ,...,T ( w n )) spans R ( T ). Solution: Suppose that w R ( T ). Then there exists v V such that T ( v ) = w . We can write v = a 1 u 1 + ··· + a m u m + b 1 w 1 + ··· + b n w n , because ( u 1 ,..., u m , w 1 ,..., w n ) is a basis for V , and then T ( v ) = a 1 T ( u 1 ) + ··· + a m T ( u m ) + b 1 T ( w 1 ) + ··· + b n T ( w n ) , by linearity of T . Since u 1 ,..., u m lie in N ( T ), w = T ( v ) = b 1 T ( w 1 ) + ··· + b n T ( w n ) . Thus w lies in the span of ( T ( w 1 )
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Unformatted text preview: ,...,T ( w n )). 2. Prove that the list ( T ( w 1 ) ,...,T ( w n )) is linearly independent. Solution: Suppose that b 1 T ( w 1 ) + + b n T ( w n ) = , 1 for some elements b 1 ,...,b n of the eld. Then T ( b 1 w 1 + + b n w n ) = , so b 1 w 1 + + b n w n N ( T ) . But then b 1 w 1 + + b n w n = a 1 u 1 + + a m u m , for some choice of scalars a 1 ,...,a m , since ( u 1 ,..., u m ) is a basis for N ( T ). Thus a 1 u 1 + + a m u m-b 1 w 1--b n w n = . Since the list ( u 1 ,..., u m , w 1 ,..., w n ) is linearly independent, a 1 = = a m = b 1 = = b n = 0 . From this we conclude that b 1 = = b n = 0 , and thus ( T ( w 1 ) ,...,T ( w n )) is linearly independent. 2...
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108AQuiz2 - ,...,T ( w n )). 2. Prove that the list ( T ( w...

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