115_solns_5

# 115_solns_5 - 115A HW 5 Sec 2.3. 11. Let V be a vector...

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Unformatted text preview: 115A HW 5 Sec 2.3. 11. Let V be a vector space and let T : V V be a linear operator on V . Show that T 2 = T (that is, T 2 is the zero transformation) if and only if R ( T ) N ( T ). Suppose T 2 = T . Let u R ( T ); that is, there exists v V such that u = T ( v ). Since T 2 = T , we have T 2 ( v ) = 0, so T ( u ) = T ( T ( v )) = 0. In other words, u N ( T ), and so R ( T ) N ( T ). Conversely, suppose R ( T ) N ( T ). Let v V and u = T ( v ). Since u R ( T ), we have u N ( T ). So, T ( u ) = 0 and T 2 ( v ) = T (0) = 0. Since v was arbitrary, we conclude that T 2 = T . 13. Let A = ( a ij ) and B = ( b ij ) be n n matrices. Recall that tr( A ) = n X i =1 a ii . Prove that tr( AB ) = tr( BA ) and tr( A ) = tr( A t ). The latter statement is immediate from the definition of A t (namely, the diagonal entries of A and A t are identical). To prove the former, let C = AB = ( c ij ) and D = BA = ( d ij ). By definition of matrix multiplication, we have)....
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## This note was uploaded on 10/20/2010 for the course MATH MAT443 taught by Professor Jones during the Fall '09 term at ASU.

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115_solns_5 - 115A HW 5 Sec 2.3. 11. Let V be a vector...

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