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 k1 A k1 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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 '07
 Guralnick
 Math, Differential Equations, Linear Algebra, Algebra, Equations, Vector Space, #, scalar multiplication, 0 %, ≤F VF, $ 1 1 3

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