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3.4 Systems of Linear Eq.

# 3.4 Systems of Linear Eq. - 3.4 3.4 Systems of Linear...

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3.4. 3.4. Systems of Linear equations - Computational aspects 1. (a) F (a finite row operations) (b) T (P.182 Corollary) (c) T (P.158 Corollary 1 to Theorem 3.6) (d) T (p.187 Theorem 3.14) (e) F The system has a solution if and if only the echelon form of the augmented matrix M does not have a row of the form (0 , · · · , 0 , b ) with b 6 = 0 (f) T rank ( A ) = rank ( A | b ) ( A | b ) is consistent If the system ( A | b ) is consistent and rank ( A ) = r , then the dimension of the solution set is n - r . (g) T Since A is row equivalent to A 0 i.e.A 0 = EA rank ( A ) = rank ( EA 0 ) = rank ( A 0 ) 2. (a) 4 - 3 1 (b) 9 4 0 + r - 5 - 3 1 | r R 178 PNU-MATH

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3.4. (c) 2 3 - 2 1 (d) 13 22 - 1 26 18 13 + r 9 - 15 0 1 | r R (e) 4 0 1 0 + r 4 1 0 0 + s 1 0 2 1 | r, s R (f) - 3 3 1 0 + r 1 - 2 0 1 | r R (g) - 23 0 7 9 0 + r 1 1 0 0 0 + s - 23 0 6 9 1 | r, s R 3. (a) ( ) If ( A 0 | b 0 ) has such a row, say k - th row, then the corresponding k - th equation in the system A 0 x = b 0 is 0 x 1 + 0 x 2 + · · · + 0 x n = c k , c k 6 = 0 ; which has no solutions i.e. A 0 x = b 0 is inconsistent rank ( A 0 ) 6 = rank ( A 0 | b 0 ) ( ) Assume that ( A 0 | b 0 ) has no such a row, then b 0 R ( L A 0 ) 179 PNU-MATH
3.4.

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