Page 836 - Advanced_Engineering_Mathematics o'neil

816 Answers to Selected Problems

9. Not collinear, 29x + 37y − 12z = 30
11. i − j + 2k
13. F × G = F G sin(θ), and this is the area of a parallelogram having incident sides of length F and G and
incident angle θ.

Section 6.4 The Vector Space R n
1. Independent 3. Independent 5. Dependent
7. Dependent 9. Independent
11. A basis consists of < 1,0,0,−1 > and < 0,1,−1,0 > and the dimension is 2.
13. A basis consists of < 1,0,0,0 >,< 0,0,1,0 > and < 0,0,0,1 >. The dimension is 3.
15. The vector < 0,1,0,2,0,3,0 > forms a basis and the dimension is 1.
17. 1 1
X = < 1,1,1,1,0 > + < −1,1,0,0,0 >
4 2
11
− < 1,1,−1,−1,0 > + < 0,0,2,−2,0 > −2 < 0,0,0,0,2 >
4
19. Since a basis spans the space, form some numbers a 1 ,··· ,c k ,
U = a 1 V 1 + a 2 V 2 + ··· + a k V k .
If U
= O, some a j
= 0, so
U − a 1 V 1 −··· − a k V k = O,
and therefore, U,V 1 ,··· ,V k are linearly dependent by Theorem 6.1(1). If U = O,then

U − 0V 1 −··· − 0V k
again shows by Theorem 6.1(1) that U,V 1 ,··· ,V k are linearly dependent.
21. If X · X = Y · Y,then
(X − Y) · (X + Y) = X · X + X · Y + Y · X − Y · Y = 0.
n
23. Write X = j=1 (X · V j )V j .Then
2
X = X · X
n n

= (X · V j )V j · (X · V k )V k
j=1 k=1
n n

= (X · V j )(X · V k )
j=1 k=1
n

2
= (X · V j ) .
j=1
Section 6.5 Orthogonalization
1. V 1 =< 1,4,0 >,V 2 =< 52/17,−13/17,0 >
3. V 1 =< 0,2,1,1 >,V 2 =< 0,4/3,13/6,29/6 >,V 3 =< 0,7/179,−11/179,3/179 >
5. V 1 =< 0,0,2,2,1 >,V 2 =< 0,0,−1/9,−19/9,40/9 >,
V 3 =< 0,1,−341/218,279/218,62/218 >,V 4 =< 0,248/393,88/393,−24/131,−32/393 >
7. V 1 =< 0,0,1,1,0,0 >,V 2 =< 0,0,−3/2,3/2,0,0 >

Section 6.6 Orthogonal Complements and Projections
1. u S =< −2,6,0,0 >,u = u − u S =< 0,0,1,7 >
⊥
3. u S =< 9/2,−1/2,0,5/2,−13/2 >,u =< −1/2,−1/2,3,−1/2,−1/2 >
⊥
5. u S =< 3,1/2,3,1/2,3,0,0 >,u =< 5,1/2,−2,−1/2,−3,−3,4 >
⊥
n
⊥
7. If u 1 ,··· ,u k is a basis for S and v 1 ,··· ,v m a basis for S , then any vector w in R has a unique representation
w = c 1 u 1 +··· + c k u k + d 1 v 1 + ··· + d m v m .

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October 14, 2010 17:50 THM/NEIL Page-816 27410_25_Ans_p801-866