chapter02

# Linear Algebra with Applications (3rd Edition)

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Chapter 2 SSM: Linear Algebra Chapter 2 2.1 1. Not a linear transformation, since y 2 = x 2 + 2 is not linear in our sense. 3. Not linear, since y 2 = x 1 x 3 is nonlinear. 5. By Fact 2.1.2, the three columns of the 2 × 3 matrix A are T ( ~ e 1 ) , T ( ~ e 2 ), and T ( ~ e 3 ), so that A = 7 6 - 13 11 9 17 . 7. Note that x 1 ~v 1 + · · · + x m ~v m = [ ~v 1 . . .~v m ] x 1 · · · x m , so that T is indeed linear, with matrix [ ~v 1 ~v 2 · · · ~v m ]. 9. We have to attempt to solve the equation y 1 y 2 = 2 3 6 9 x 1 x 2 for x 1 and x 2 . Reducing the system 2 x 1 + 3 x 2 = y 1 6 x 1 + 9 x 2 = y 2 we obtain x 1 + 1 . 5 x 2 = 0 . 5 y 1 0 = - 3 y 1 + y 2 . No unique solution ( x 1 , x 2 ) can be found for a given ( y 1 , y 2 ); the matrix is noninvertible. 11. We have to attempt to solve the equation y 1 y 2 = 1 2 3 9 x 1 x 2 for x 1 and x 2 . Reducing the system x 1 + 2 x 2 = y 1 3 x 1 + 9 x 2 = y 2 we find that x 1 = 3 y 1 - 2 3 y 2 x 2 = - y 1 + 1 3 y 2 . The inverse matrix is 3 - 2 3 - 1 1 3 . 13. a. First suppose that a 6 = 0. We have to attempt to solve the equation y 1 y 2 = a b c d x 1 x 2 for x 1 and x 2 . ax 1 + bx 2 = y 1 cx 1 + dx 2 = y 2 ÷ a x 1 + b a x 2 = 1 a y 1 cx 1 + dx 2 = y 2 - c ( I ) x 1 + b a x 2 = 1 a y 1 ( d - bc a ) x 2 = - c a y 1 + y 2 28

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Chapter 2 SSM: Linear Algebra 15. By Exercise 13a, the matrix a - b b a is invertible if (and only if) a 2 + b 2 6 = 0, which is the case unless a = b = 0. If a - b b a is invertible, then its inverse is 1 a 2 + b 2 a b - b a , by Exercise 13b. 17. If A = - 1 0 0 - 1 , then A~x = - ~x for all ~x in R 2 , so that A represents a reflection about the origin. This transformation is its own inverse: A - 1 = A . 19. If A = 1 0 0 0 , then A x 1 x 2 = x 1 0 , so that A represents the orthogonal projection onto the ~ e 1 axis. (See Figure 2.1.) This transformation is not invertible, since the equation A~x = 1 0 has infinitely many solutions ~x . Figure 2.1: for Problem 2.1.19 . 21. Compare with Example 5.

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• Spring '08
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