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**Unformatted text preview: **+ a 1 + a = 0} R R [ x ] R . G. If A M n ( R ) and B R n , show that { c k A k B + c k-1 A k-1 B + + c 1 AB + c B R n | k 0 and all c i R } is a subspace of R n . H. Let W = { A M n ( R ) | A ik k = n " = A jk k = n " for all 1 i , j n }. Is W R M n ( R )? I. If A M n ( R ) show that A 2 = 0 nxn Im( A ) null space( A ). J. If A M n ( R ) and if A 2 = A then R n = Im( A ) + Im( A I n ) and Im( A ) Im( A I n ) = {0 nxn }. (See C and D above.) K. Let A = 2 1 1 3 1 " 1 1 1 3 " 1 1 1 " 2 " 2 # $ % % % & ' ( ( ( . Describe explicitly the vectors in Im( A ) and the vectors in null space( A ) (as in Chapter 2). What is the intersection of these two subspaces? Hand In 5A) pp. 249 250 #6; 12; B; C; D; pp. 257 258 #2; 10; 12; F; K 5B) E; G; I; J; Due Wednesday September 28...

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