MATH535_HW12 - 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

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Unformatted text preview: 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 ....
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This note was uploaded on 04/01/2008 for the course MATH 535 taught by Professor Brownawell,woodro during the Fall '08 term at Pennsylvania State University, University Park.

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MATH535_HW12 - 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

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