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# HW 5 - a 1 a = 0 ≤ R R x R G If A ∈ M n R and B ∈ R n...

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MATH 225 HOMEWORK 5 pp. 249 – 250 T/F Review, and Problems #2 – 4, 6, 11 – 13, 17 – 19 pp. 257 – 258 #2 – 6, 8 – 13, 16, 18, 20, 21, 24. Recall that W F V F means that W is a subspace of V F . Also below M n ( F ) assumes n > 1. A. Let V = M n ( R ) and for A , B V set A # B = A T + B T . Is V R a vector space using # as the addition on V and as the usual scalar multiplication of matrices? B. Let V be the infinite sequences of rational numbers that have limit zero: V = {( a 1 , a 2 , . . . , a n , . . .) | all α j Q and lim j "# a j = 0}. Define + and as: ( a 1 , a 2 , . . . , a n , . . .) + ( b 1 , b 2 , . . . , b n , . . .) = ( a 1 + b 1 , a 2 + b 2 , . . . , a n + b n , . . .) and c ( a 1 , a 2 , . . . , a n , . . .) = ( ca 1 , ca 2 , . . . , ca n , . . .). Is V Q a vector space? C. Let W 1 , . . . , W k F V F and show that W i " = { 1 + + k V | each j W j } F V F . D. If W 1 , . . . , W k F V F show that W j ! F V F . E. If W , U F V F , show that W U = V W = V or U = V . (To show this consider what results if the conclusion is false.) F. Show that { a n x n + + a 1 x + a 0 R [ x ] | n 0 and a n +
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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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