Math416_HW3_sol - Math 416 - Abstract Linear Algebra Fall...

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Unformatted text preview: 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 =...
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This note was uploaded on 11/07/2011 for the course MATH 416 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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Math416_HW3_sol - Math 416 - Abstract Linear Algebra Fall...

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