Page 194 - Advanced engineering mathematics

174 CHAPTER 6 Vectors and Vector Spaces

EXAMPLE 6.18
The vectors
v 1 =< 2,0,1,0,0 >,v 2 =< 0,5,0,0,0 >,v 3 =< −1,0,2,0,0 >
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form an orthogonal basis for a three-dimensional subspace S of R .Let X =< 12,−5,4,0,0 >,
a vector in S. We will find the coordinates c 1 ,c 2 ,c 3 of X with respect to this basis. Compute
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X · v 1
c 1 = = ,
v 1 2 5
−25
X · v 2
c 2 = = =−1,
v 2 2 25
4
X · v 3
c 3 = =− .
v 3 2 5
Then
X = c 1 v 1 + c 2 v 2 + c 3 v 3 .

We will pursue further properties of sets of orthogonal vectors in the next section.

SECTION 6.4 PROBLEMS

6
In each of Problems 1 through 10, determine whether 14. S consists of all vectors in R of the form < x, x,
the vectors are linearly independent or dependent in the y, y,0, z >.
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appropriate R . 7
15. S consists of all vectors <0, x,0,2x,0,3x,0> in R .
1. 3i + 2j,i − j in R 3 In each of Problems 16, 17, and 18, find the coordinates of
2. 2i,3j,5i − 12k,i + j + k in R 3 X with respect to the given basis.
3. < 8,0,2,0,0,0,0 >,< 0,0,0,0,1,−1,0 > in R 7 16. X =< 4,4,−1,2,0 > with vectors < 2,1,0,0,0 >,
4. < 1,0,0,0 >,< 0,1,1,0 >,< −4,6,6,0 > in R 4 < 1,−2,0,0,0 >, < 0,0,3,−2,0 >, < 0,0,2,
5
−3,0 > spanning a subspace S of R .
5. < 1,2,−3,1 >,< 4,0,0,2 >,< 6,4,−6,4 > in R 4
17. X =< −3,−2,5,1,−4 >, with the basis < 1,1,1,
6. < 0,1,1,1 >,< −3,2,4,4 >,< −2,2,34,2 >, 1,0 >, < −1,1,0,0,0 >, < 1,1,−1,−1,0 >, < 0,0,
< 1,1,−6,−2 > in R 4
5
2,−2,0 >, < 0,0,0,0,2 > of R .
7. < 1,−2 >,< 4,1 >,< 6,6 > in R 2
18. X =< −3,1,1,6,4,5 >, with the basis < 4,0,1,
8. < −1,1,0,0,0 >,< 0,−1,1,0,0 >,< 0,1,1,1,0 > 0,0,0 >, < −1,1,4,0,0,0 >, < 0,0,0,2,1,0 >,
in R 5 < 0,0,0,−1,2,5 >, < 0,0,0,0,0,5 >.
9. < −2,0,0,1,1 >,< 1,0,0,0,0 >,< 0,0,0,0,2 >, 19. Suppose V 1 ,··· ,V k form a basis for a subspace S of
< 1,−1,3,3,1 > in R 5 R .Let U be any other vector in S. Show that the
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10. < 3,0,0,4 >,< 2,0,0,8 > in R 4 vectors V 1 ,··· ,V k , U are linearly dependent.
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20. Let V 1 ,··· ,V k be mutually orthogonal vectors in R .
In each of Problems 11 through 15, show that the set S is Prove that
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a subspace of the appropriate R and find a basis for this
2
2
2
subspace and the dimension of the subspace. V 1 +···+ V k = V 1 +···+ V k .
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11. S consists of all vectors < x, y,−y,−x > in R . Hint: Write
2
4
12. S consists of all vectors < x, y,2x,3y > in R . V 1 +···+V k = (V 1 +···+V k )·(V 1 +···+V k ).
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13. S consists of all vectors in R with zero second 21. Let X and Y be vectors in R , and suppose that X =
component. Y . Show that X − Y and X + Y are orthogonal.
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