Adv Alegbra HW Solutions 131

Adv Alegbra HW Solutions 131 - 131 (ii) Prove that lies in...

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131 (ii) Prove that β lies in the column space of A if and only if rank ( [ A | β ] ) = rank ( A ) . Solution. Let A be Gaussian equivalent to an echelon matrix U , so that there is a nonsingular matrix P with PA = U . Then β lies in the row space Ro w( A ) if and only if P β U ) . (iii) Does β = ( 0 , 3 , 5 ) lie in the subspace spanned by α 1 = ( 0 , 2 , 3 ) , α 2 = ( 0 , 4 , 6 ) , α 3 = ( 1 , 1 , 1 ) ? Solution. No. 4.30 (i) Prove that an n × n matrix A over a f eld k is nonsingular if and only if it is Gaussian equivalent to the identity I . Solution. Absent. (ii) Find the inverse of A = 231 110 101 . Solution. If E p ··· E 1 A = I , then A 1 = E 1 1 E 1 p . Con- clude that the elementary row operations which change A into I also change I into A 1 . The answer is A 1 = 1 4 1 3 1 11 1 13 5 . 4.31 (i) Let Ax = b be an m × n linear system over a f eld k . Prove that there exists a solution x = ( x 1 ,..., x n ) with
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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