선형대수학 3판 솔루션 (Linear Algebra and Its applications 3th)

1.1

SOLUTIONS

Notes: The key exercises are 7 (or 11 or 12), 19?22, and 25. For brevity, the symbols R1, R2,…, stand for row 1 (or equation 1), row 2 (or equation 2), and so on. Additional notes are at the end of the section.
1.

x1 + 5 x2 = 7 ?2 x1 ? 7 x2 = ?5

? 1 ? ?2 ?

5 ?7

7? ?5? ?
x1 + 5 x2 = 7

Replace R2 by R2 + (2)R1 and obtain:

3x2 = 9
x1 + 5 x2 = 7 x2 = 3 x1

?1 ?0 ? ?1 ?0 ? ?1 ?0 ?

5 3 5 1 0 1

7? 9? ? 7? 3? ? ?8? 3? ?

Scale R2 by 1/3: Replace R1 by R1 + (?5)R2: The solution is (x1, x2) = (?8, 3), or simply (?8, 3).
2.

= ?8
x2 = 3

2 x1 + 4 x2 = ?4 5 x1 + 7 x2 = 11

?2 ?5 ?

4 7

?4 ? 11 ? ?
x1 + 2 x2 = ?2

Scale R1 by 1/2 and obtain: Replace R2 by R2 + (?5)R1: Scale R2 by ?1/3: Replace R1 by R1 + (?2)R2: The solution is (x1, x2) = (12, ?7), or simply (12, ?7).

5 x1 + 7 x2 = 11
x1 + 2 x2 = ?2

?1 ?5 ? ?1 ?0 ? ?1 ?0 ? ?1 ?0 ?

2 7 2 ?3 2 1 0 1

?2 ? 11 ? ? ?2 ? 21? ? ?2 ? ?7 ? ? 12 ? ?7 ? ?

?3x2 = 21
x1 + 2 x2 =