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Ans-Lin-Alg - Part I Linear Algebra Exercises 1.1 1 3 6 8...

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Part I. Linear Algebra Exercises 1.1 1. x = 4 , y = 1 3. The set of points on the line x + 2 y = 4 6. x = 2 , y = - 1 8. x = - 2 , y = 3 12. x = - 3 + 2 y, z = 0 , y any real number 15. x = 2 , y = 1 , z = 1 Exercises 1.2 3. Coefficient matrix: rank 2; augmented matrix: rank 2; x = 4 - 2 a, y = a, z = - 2 , a any real number 4. Coefficient matrix: rank 2; augmented matrix: rank 3; no solution 7. Coefficient matrix: rank 3; augmented matrix: rank 3; x 1 = 8 + 2 a - 3 b, x 2 = a, x 3 = - 1 - 2 b, x 4 = b, x 5 = - 3 , a, b any real numbers 13. x 1 = 10 / 7 , x 2 = 2 / 7 , x 3 = 3 / 2 19. x 1 = 3 - 2 a, x 2 = a, x 3 = 2 , x 4 = 1 , a any real number 21. no solution 23. ( i ) k = - 3 , 2 ( ii ) k = - 3 ( iii ) k = 2 25. (a) no (b) no (c) yes 26. 1 2 4 - 1 3 0 1 3 2 2 2 - 2 2 3 4 3 4 1 1 0 0 11 / 3 1 8 / 3 0 1 0 1 / 3 0 - 4 / 3 0 0 1 - 4 / 3 5 1 / 6 Exercises 1.3 3. yes 1
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4. yes 5. No; the leading 1 in the 3,5-position is not the only nonzero entry in its column. 6. 8. yes 11. x = - 3 - a, y = 2 + 2 a, z = a, a any real number 13. x 1 = 11 - 2 a + b, x 2 = a, x 3 = 3 - b, x 4 = b, a, b any real numbers 18. x = y = z = 0, the trivial solution 21. x 1 = 2 a - b, x 2 = - a + 4 b, x 3 = a, x 4 = b, a, b any real numbers 28. a = b b any real number 29. a = 4 Exercises 1.4
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