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Math416_HW3_sol - Math 416 Abstract Linear Algebra Fall...

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Math 416 - Abstract Linear Algebra Fall 2011, section E1 Homework 3 solutions Section 1.6 6.1. (2 pts) We want to show that any w W can be expressed uniquely as a linear combination of Av 1 , . . . , Av n . The equation w = c 1 Av 1 + . . . + c n Av n = A ( c 1 v 1 + . . . + c n v n ) is equivalent to the equation A - 1 w = A - 1 A ( c 1 v 1 + . . . + c n v n ) = c 1 v 1 + . . . + c n v n which has a unique solution ( c 1 , . . . , c n ) since { v 1 , . . . , v n } is a basis of V . 6.2. (1 pt check) A right inverse to A = 1 1 is a 2 × 1 matrix B = c d satisfying AB = I 1 , that is AB = 1 1 c d = c + d = 1 . The right inverses of A are all matrices of the form c 1 - c for some c R . In particular, A has distinct right inverses, therefore A has no left inverse (by thm 6.1). 6.8. A cannot be invertible. If A were invertible, the condition AB = 0 would imply A - 1 AB = A - 1 0 = 0, that is B = 0. 6.9. (2 pts) T 1 x 1 x 2 x 3 x 4 x 5 = x 1 x 4 x 3 x 2 x 5 = 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 x 1 x 2 x 3 x 4 x 5 T 2 x 1 x 2 x 3 x 4 x 5 = x 1 x 2 + ax 4 x 3 x 4 x 5 = 1 0 0 0 0 0 1 0 a 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 x 1 x 2 x 3 x 4 x 5
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