Page 240 - Advanced engineering mathematics

220 CHAPTER 7 Matrices and Linear Systems

7. −10x 1 − x 2 + 4x 3 − x 4 + x 5 − x 6 = 0 12. 2x 1 − 4x 5 + x 7 + x 8 = 0
x 2 − x 3 + 3x 4 = 0 2x 2 − x 6 + x 7 − x 8 = 0
2x 1 − x 2 + x 5 = 0 x 3 − 4x 4 + x 5 = 0
x 2 − x 4 + x 6 = 0 x 2 − x 3 + x 4 = 0
x 2 − x 5 + x 6 − x 7 = 0
8. 8x 1 − 2x 3 + x 6 = 0
2x 1 − x 2 + 3x 4 − x 6 = 0
13. Can a system AX = O having at least as many equa-
x 2 + x 3 − 2x 5 − x 6 = 0
tions as unknowns, have a nontrivial solution?
x 4 − 3x 5 + 2x 6 = 0
9. x 2 − 3x 4 + x 5 = 0 14. Show that a system AX = O has a nontrivial solu-
2x 1 − x 2 + x 4 = 0 tion if and only if the columns of A are linearly
2x 1 − 3x 2 + 4x 5 = 0 dependent. Hint: This can be done using a dimension
argument. Another approach is to write AX as a lin-
10. 4x 1 − 3x 2 + x 4 + x 5 − 3x 6 = 0
ear combination of the columns of A, as suggested in
2x 2 + 4x 4 − x 5 − 6x 6 = 0
Section 7.1.1.
3x 1 − 2x 2 + 4x 5 − x 6 = 0
2x 1 + x 2 − 3x 3 + 4x 4 = 0
15. Let A be an n × m matrix of real numbers. Let S(A)
11. x 1 − 2x 2 + x 5 − x 6 + x 7 = 0 denote the solution space of A.Let R be the row space
x 3 − x 4 + x 5 − 2x 6 + 3x 7 = 0 and C the column space of A.
⊥
x 1 − x 5 + 2x 6 = 0 (a) Show that R = S(A).
t
⊥
2x 1 − 3x 4 + x 5 = 0 (b) Show that C = S(A ).
7.6 Nonhomogeneous Systems

Now consider the nonhomogeneous linear system of n equations in m unknowns:

a 11 x 1 + a 12 x 2 + ··· + a 1m x m = b 1
a 21 x 1 + a 22 x 2 + ··· + a 2m x m = b 2
.
.
.
a n1 x 1 + a n2 x 2 + ··· + a nm x m = b n .
In matrix form,
AX = B (7.1)
where A is the coefficient matrix,
⎛ ⎞ ⎛ ⎞
x 1 b 1
x 2 b 2
⎜ ⎟ ⎜ ⎟
X = ⎜ . ⎟ and B = ⎜ . ⎟.
⎜
⎟
⎜ ⎟
.
.
⎝ . ⎠ ⎝ . ⎠
x m b n
The system is nonhomogeneous if at least one b j = 0. Nonhomogeneous systems differ
from linear systems in two significant ways.
1. A nonhomogeneous system may have no solution. For example, the system
2x 1 − 3x 2 = 6
4x 1 − 6x 2 = 8
can have no solution. If 2x 1 − 3x 2 = 6, then 4x 1 − 6x 2 must equal 12, not 8.

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October 14, 2010 14:23 THM/NEIL Page-220 27410_07_ch07_p187-246