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Unformatted text preview: ASSIGNMENT 2 Solutions Let A be a 3 × 3 matrix. 1. [4 marks] Prove that the rows of A span R 3 if, and only if, the columns of A span R 3 . Suppose B and C are row equivalent matrices. Since each may be derived from the other via elementary row operations, the span of the rows of B is equal to the span of the rows of C . Let R denote the reduced row-echelon form of A , and I denote the 3 × 3 identity matrix. Let the rows of A span R 3 . Suppose R 6 = I ; that is, suppose R has at least one zero row — let R = r 1 r 2 r 3 r 4 r 5 r 6 . Then the span of the rows of A is equal to span r 1 r 2 r 3 , r 4 r 5 r 6 . This defines either a point, a line, or a plane. In all cases we have a contradiction; hence R = I , and the columns of A span R 3 . Let the columns of A span R 3 . Then R = I , and the span of the rows of A is equal to the span of the rows of R — that is, to R 3 . Let A = - 2 a 1 b c 3 and B = - 3 1- 8 5- 2 1 2- 2 , where a , b and c are constants....
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This note was uploaded on 04/07/2010 for the course MATH 136 taught by Professor All during the Fall '08 term at Waterloo.
- Fall '08