HW 2(1)

HW 2(1) - = 0. F. Let L = { A M n ( R ) | all row sums of A...

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MATH 225 HOMEWORK 2 p. 138 True/False #1 – 6; pp. 138 – 139 #1, 4, 6 – 11 p. 149 True/False # 4 – 9; pp. 149 – 150 #16 – 25 (Find the reduced row echelon form and rank.) Below, R is the field of real numbers. A. Let A , S , T M n ( R ) and B R n . If ST = I n show that if C R n is a solution of AX = B then TC is a solution of ( AS ) X = B . B. If A , S M n ( R ) with SA = I n = AS show that for any B R n the system AX = B is consistent and has a unique solution. C. If for some A M n ( R ) and any B R n , AX = B has a unique solution, show that AS = I n for some S M n ( R ). (Look at the solutions of AX = col j ( I n ) and think about matrix multiplication.) D. Given A M n ( R ) and r R , show that the solutions in R n of AX = rX are the solutions of some homogeneous system of linear equations (note that X = I n X ). E. Let S = { α 1 , . . . , k R 1x n }. Describe a homogeneous system of linear equations whose solutions are exactly those β R n satisfying 1 = 0, 2 = 0, . . . , k
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Unformatted text preview: = 0. F. Let L = { A M n ( R ) | all row sums of A are 1, that is A i 1 + A i 2 + + A in = 1 for all i }. Show that A L [1, 1, . . . , 1] T R n is a solution of AX = X . Use this to show: A , B L AB L . G. Show that A M n ( R ) has rank n the row reduced echelon form of A is I n . H. If A M n ( R ) has rank n then the systems AX = B is consistent for any B R n (Use G. ) I. If A , B R m x n have the same row reduced echelon form E , then show that the homogeneous systems AX = 0 n x1 and BX = n x1 are equivalent. J. Describe those a b c " # $ $ % & ' ' R 3 so that the system 1 1 1 2 1 1 7 2 " # $ $ % & ' ' X = a b c " # $ $ % & ' ' is consistent. Hand In 2A) p. 139 #8; 10; 11; B; pp. 149 150 #18; 23; 24; F 2B) C; G; H; J Due Wednesday September 7...
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This note was uploaded on 12/16/2011 for the course MATH 225 at USC.

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