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# Sol6 - SOLUTION TO ASSIGNMENT 6 MATH 251 WINTER 2007 1 Let...

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SOLUTION TO ASSIGNMENT 6 - MATH 251, WINTER 2007 1. Let T : V W be a linear map and define T * : W * V * by ( T * ( g ))( v ) := g ( Tv ). Prove the following lemma: L EMMA 1. (1) T * is a well-defined linear map. (2) Let B, C be bases to V, W , respectively. Let A = C [ T ] B be the m × n matrix representing T , where n = dim( V ) , m = dim( W ) . Then the matrix representing T * with respect to the dual bases B * , C * is the transpose of A : B * [ T * ] C * = C [ T ] B t . (3) If T is injective then T * is surjective. (Do NOT use Proposition 7.2.8 in the notes). (4) If T is surjective then T * is injective. Proof. (1) First, we need to show that the map v g ( T ( v )) is a linear map. This is easy: T and g are linear maps and so is their composition. Next, we need to check that T * is linear. Namely, that for every v , a i F , g i W * , we have ( T * ( a 1 g 1 + a 2 g 2 ))( v ) = ( a 1 T * g 1 + a 2 T * g 2 )( v ) . This follows from definitions: ( T * ( a 1 g 1 + a 2 g 2 ))( v ) = ( a 1 g 1 + a 2 g 2 )( Tv ) = a 1 · g 1 ( Tv )+ a 2 · g 2 ( Tv ) = a 1 · T * g 1 ( v ) + a 2 · T * g 2 ( v ) = ( a 1 T * g 1 + a 2 T * g 2 )( v ). (2) Let us write B = { b 1 , . . . , b n } , C = { c 1 , . . . , c m } , B * = { b * 1 , . . . , b * n } , C * = { c * 1 , . . . , c * m } , where we have b * i ( b j ) = δ ij , c * i ( c j ) = δ ij . If C [ T ] B = A = ( a ij ) that means that [ Tb i ] C = a 1 i c 1 + · · · + a mi c m . Let us write T * c * i = α 1 i b * 1 + · · · + α ni b * n , so that B * [ T * ] C * = ( α ij ) , and we wish to prove that α ji = a ij . As we have seen before, the coefficient α ji is equal to ( T * c * i )( b j ) (cf. the solution to question (2) below; we make use of B being the dual basis to B * . See also the notes, Example 7.1.4). But, by definition, ( T * c * i )( b j ) = c * i ( Tb j ) = c * i ( a 1 j c 1 + · · · + a mj c m ) = a ij . (3) Suppose T : V W is injective. This means that the column rank of A = C [ T ] B is maximal, equal to n , which is the row rank of A t . But then also the column rank of A t is n . Therefore, the image of T * is n -dimensional. Since dim(

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Sol6 - SOLUTION TO ASSIGNMENT 6 MATH 251 WINTER 2007 1 Let...

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