Page 260 - Advanced engineering mathematics

240 CHAPTER 7 Matrices and Linear Systems

Because U is upper triangular, this is the system

4x 1 + 3x 2 + 3x 3 − 4x 4 + 6x 5 = 4,
1 7 5
x 2 − x 3 + 4x 4 + x 5 =−3
4 4 2
−2x 3 − 7x 5 = 10
575
140x 4 + x 5 =−408.
2
Solve this to obtain the solution of the solution of the original system:

⎛ ⎞ ⎛ ⎞
523/28 3971/140
⎜ −183/7 ⎟ ⎜ −1238/35 ⎟
⎜ ⎟ ⎜ ⎟
X = α ⎜ −7/2 ⎟ + ⎜ −5 ⎟ .
⎜ ⎟ ⎜ ⎟
⎝ −115/56 ⎠ ⎝ −102/35 ⎠
1 0

SECTION 7.9 PROBLEMS

In each of Problems 1 through 6, find an LU factorization In each of Problems 7 through 12, solve the system AX=B
of the matrix. by factoring A. A is given first, then B
⎛ ⎞
2 4 −6 ⎛ ⎞ ⎛ ⎞
4 4 2 1
1. ⎝ 8 2 1 ⎠
7. ⎝ 1 −1
−4 4 10 3 ⎠ , ⎝ 0 ⎠
1 42 2 1
⎛ ⎞
1 5 2

2. ⎝ 3 −4 2 ⎠ 2 1 1 3 2
8. ,
1 4 10 1 4 6 2 4
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
−2 1 12 −1 1 1 6 2
3. ⎝ 2 −6 1 ⎠ 9. ⎝ 2 1 0 4 ⎠ , ⎝ 1 ⎠
2 2 4 1 −2 4 6 6
⎛ ⎞
1 7 2 −1 ⎛ 7 2 −4 ⎞ ⎛ 7 ⎞
4. ⎝ 3 5 2 6 ⎠ 10. ⎝ −3 2
−3 −7 10 −4 8 ⎠ , ⎝ −1 ⎠
4 4 20 3
⎛ ⎞
1 4 2 −1 4
⎛ ⎞ ⎛ ⎞
1 −1 4 −1 4 6 1 −1 3 4
⎜ ⎟
5. ⎜ ⎟
⎝ −2 6 8 6 −2 ⎠ 11. ⎜ 4 2 1 5 ⎟ ⎜ 12 ⎟
,
⎟ ⎜
⎜
⎟
4 2 1 2 −4 ⎝ −4 1 6 5 ⎠ ⎝ 2 ⎠
2 −1 −1 4 −3
⎛ ⎞
4 −8 2
2 24 −2 ⎛ 1 2 0 1 1 2 −4 ⎞ ⎛ ⎞
⎜ ⎟ 0
6. ⎜ ⎟
⎝ −3 2 14 ⎠ 12. ⎝ 3 3 −3 6 −5 2 5 ⎠ , ⎝ −4 ⎠
0 1 −5 6 8 4 0 −2 2 0 2
7.10 Linear Transformations
n
m
Sometimes we want to consider functions between R and R . Such a function associates with
m
n
each vector in R a vector in R , according to a rule defined by the function.
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October 14, 2010 14:23 THM/NEIL Page-240 27410_07_ch07_p187-246