# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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366

Chapter 7. Systems of First Order Linear Equations

3. If x(1) and x(2) are eigenvectors that correspond to different eigenvalues, then (x( , x(2)) = 0. Thus, if all eigenvalues are simple, then the associated eigenvectors form an orthogonal set of vectors.

4. Corresponding to an eigenvalue of multiplicity m, it is possible to choose m eigenvectors that are mutually orthogonal. Thus the full set of n eigenvectors can always be chosen to be orthogonal as well as linearly independent.

Example 5 above involves a real symmetric matrix and illustrates properties 1, 2, and 3, but the choice we have made for x(2) and x(3) does not illustrate property 4. However, it is always possible to choose an x(2) and x(3) so that (x(2), x(3)) = 0. For example, in Example 5 we could have chosen

as the eigenvectors associated with the eigenvalue ê = —1. These eigenvectors are orthogonal to each other as well as to the eigenvector x(1) corresponding to the eigenvalue ê = 2. The proofs of statements 1 and 3 above are outlined in Problems 32 and 33.

PROBLEMS In each of Problems 1 through 5 either solve the given set of equations, or else show that there is no solution.

1. x1 --- x3 =0 2. x1 --- x3 =1

x

2

+

3 x1 + x2 + x3 =1 2x1 + x2 + x3 =1

--- x1 + x2 + 2x3 =2 x1 --- x2 + 2x3 =1

3. x1 + 2x2 --- x3 =2 4. x1 --- x3 =0

x

2

+

2x1 + x2 + x3 =1 2x1 + x2 + x3 =0

x1 --- x2 + 2x3 = -1 x1 --- x2 + 2x3 =0

5. x1 — x3 = 0

3x1 + x2 + x3 = 0 —x1 + x2 + 2x3 = 0

In each of Problems 6 through 10 determine whether the given set of vectors is linearly independent. If linearly dependent, find a linear relation among them. The vectors are written as row vectors to save space, but may be considered as column vectors; that is, the transposes of the given vectors may be used instead of the vectors themselves.

6. x(1) = (1, 1, 0), x(2) = (0, 1, 1), x(3) = (1, 0, 1)

7. x(1) = (2, 1, 0), x(2) = (0, 1, 0), x(3) = (—1, 2, 0)

8. x(1) = (1, 2, 2, 3), x(2) = (—1, 0, 3, 1), x(3) = (—2, —1, 1, 0),

x(4) = (—3, 0, — 1, 3)

9. x(1) = (1, 2, —1, 0), x(2) = (2, 3, 1, —1), x(3) = (—1, 0, 2, 2),

x(4) = (3, —1, 1, 3)

10. x(1) = (1, 2, —2), x(2) = (3, 1, 0), x(3) = (2, —1, 1), x(4) = (4, 3, —2)

11. Suppose that the vectors x(1),..., x(m) each have n components, where n < m. Show that x(1), ..., x(m) are linearly dependent.

In each of Problems 12 and 13 determine whether the given set of vectors is linearly independent for —æ < t < æ. If linearly dependent, find the linear relation among them. As in Problems 6 through 10 the vectors are written as row vectors to save space.

12. x(1)(t) = (e—t, 2e—t), x(2)(t) = (e—t, e—t), x(3)(t) = (3e—t, 0)

7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 367

13. x(1)(t) = (2 sin t, sin t), x(2)(t) = (sin t, 2 sin t)

14. Let

x(')(,) = (?), x(2)(t) = (1).

Show that x(1) (t) and x(2) (t) are linearly dependent at each point in the interval 0 < t < 1. Nevertheless, show that x(1) (t) and x(2) (t) are linearly independent on 0 < t < 1.

In each of Problems 15 through 24 find all eigenvalues and eigenvectors of the given matrix.

15.

17.

19.

21.

5 -1

,3 1

-2 1

v 1 -2

' 1 V3

vV3 -1

/11/9 -2/9 8/9 >

23. I -2/9 2/9 10/9

\ 8/9 10/9 5/91

16.

18.

20.

22.

24.

3 -2

4 -1

( 1 i \-i 1

'-3 3/4

-5

f?

2 -4 -1,

Problems 25 through 29 deal with the problem of solving Ax = b when det A = 0.

25. Suppose that, for a given matrix A, there is a nonzero vector x such that Ax = 0. Show that there is also a nonzero vector y such that A*y = 0.

26. Show that (Ax, y) = (x, A*y) for any vectors x and y.

27. Suppose that det A = 0 and that Ax = b has solutions. Show that (b, y) = 0, where y is any solution of A*y = 0. Verify that this statement is true for the set of equations in Example 2.

Hint: Use the result of Problem 26.

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