MATH535_HW12 - Serge Ballif MATH 535 Homework 12 December 7...

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Serge Ballif MATH 535 Homework 12 December 7, 2007 1) Let M, N be free R -modules with bases u 1 , . . . , u m and v 1 , . . . , v n respectively. Let the transformations S : M M , T : N N , have matrices A = ( a ij ) and B = ( b kl ) with respect to these bases. (a) What is the matrix of S T with respect to the basis u i v k of M R N ? (b)Interpret the equations ( a ij ) ( b kl ) = ( a ij B ) = ( Ab kl ) sometimes found in older textbooks. (a) We know that S ( u j ) = m i =1 a ij u i , and T ( v l ) = n k =1 b kl v k . Therefore, ( S T )( u j v l ) = m X i =1 a ij u i ! n X k =1 b kl v k ! = m X i =1 n X k =1 a ij b kl ! ( u i v k ) The matrix for S T will be mn × mn , and the entry in column indexed by ( j, l ) and row indexed by ( i, k ) will be a ij b kl . (b) The equation ( a ij ) ( b kl ) = ( a ij B ) = ( Ab kl ) just expresses the fact that the above matrix has entries a ij b kl . We illustrate below. A B = a 11 B a 12 B ... a 21 B a 22 B . . . . . . = a 11 b 11 a 11 b 12 a 11 b 21 a 11 b 22 ··· a 12 b 11 a 12 b 12 a 12 b 21 a 12 b 22 ··· . . . . . . a 21 b 11 a 21 b 12 a 21 b 21 a 21 b 22 ··· . . . . . . . . . = Ab 11 Ab 12 ... Ab 21 Ab 22 . . . . . . . 2) (a) Let v be any non-zero vector from an n -dimensional vector space V over K . if u V K V , show that there are vectors v 1 = v, v 2 , . . . , v n , w 1 , . . . , w n V such that u = v 1 w 1 + · · · + v n w n . (b) Prove that, in general, such a sum cannot be shortened. That is, there exist u V K V such that there is no choice of 2 n - 2 elements v 1 , v 2 , . . . , v n - 1 , w 1 , . . . w n - 1 V for which u = v 1 w 1 + · · · + v n - 1
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