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Answers to Selected Problems 817

n
Furthermore, these vectors are linearly independent hence form a basis for R . Therefore,
dim(S) + dim(S ) = k + m = n.
⊥
9. The closest vector is
7 4 11 −11 11
u S = < 1,1,−1,0,0 > + < 0,2,1,0,0 > − < 0,1,−2,0,0 >=< ,3, , ,0 >
3 3 3 3 3

Section 6.7 The Function Space C[a,b]
x
1. V 1 (x) = e , V 2 (x) = e −x − 2 e x
e 2 −1
2
2
6
1
3. V 1 (x) = 1, V 2 (x) = x − 2/3, V 3 (x) = x − − (x − )
2 5 3
π 2 1 1
5. f S (x) = − 4cos(x) + cos(2x) − cos(3x) + cos(4x);
3 2 4
4 16 4 16
7. f S =− + cos(πx/2) − cos(πx) + cos(3πx/2)
3 π 2 π 2 9π 2
8 4 8
+ sin(πx/2) − sin(πx) + sin(3πx/2)
π π 3π
CHAPTER SEVEN MATRICES AND SYSTEMS OF LINEAR EQUATIONS
Section 7.1 Matrices
⎛ ⎞
14 −2 6
1. ⎝ 10 −5 −6 ⎠
−26 −43 −8
2 x
2 + 2x − x −12x + (1 − x)(x + e + 2cos(x))
3.
2x
x
x
4 + 2x + 2e + 2xe x −22 − 2x + e + 2e cos(x)

−36 0 68 196 20
5.
128 −40 −36 −8 72
⎛ ⎞
−10 −34 −16 −30 −14
7. AB = ⎝ 10 −2 −11 −8 −45 ⎠ , BA is not defined.
−5 1 15 61 −63
3 −18 −6 −42 66
⎛ ⎞
⎜−2 12 4 28 −44 ⎟
9. AB = (115) and BA = ⎜−6 36 12 84 −132⎟
⎟
⎜
0 0 0 0 0
⎝ ⎠
4 −24 −8 −56 88

410 36 −56 227
11. AB is not defined; BA =
17 253 40 −1

13. AB is not defined, BA = −16 −13 −5

39 −84 21
15. BA is not defined, AB =
−23 38 3
17. AB is 14 × 14, BA is 21 × 21
19. AB is not defined, BA is 4 × 2
21. AB is not defined, BA is 7 × 6
23. The number of v 1 − v 4 walks of length 3 is 4, of length 4, 18. The number of v 2 − v 3 walks of length 3 is 9. The
number of distinct v 2 − v 4 walks of length 4 is 26.
25. The number of v 4 − v 5 walks of length 2 is 2, the number of v 2 − v 3 walks of length 3 is 10, the number of v 1 − v 2
walks of length 4 is 32, the number of v 4 − v 5 walks of length 4 is 32.

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October 14, 2010 17:50 THM/NEIL Page-817 27410_25_Ans_p801-866