线性代数与解析几何--第二章 线性方程组.pdf

文档介绍：

( Systems of Linear Equations )

n
x1, · · · , xn, m
a11x1 + a12x2 + · · · + a1nxn = b1
 a21x1 + a22x2 + · · · + a2nxn = b2
.
.

 am1x1 + am2x2 + · · · + amnxn = bm



Æ
x1 = c1, · · · , xn = cn (c1 · · · cn)

℄(solution)


℄
℄Æ
℄
•

℄
•

•℄

•℄
( )
§ Gauss

Æ
℄

:



3x1 + 2x2 − x3 = 6
x1 + 3x2 + 2x3 = 9 x1 + 3x2 + 2x3 = 9

 x1 + 3x2 + 2x3 = 9
→ 2x1 − x2 + 3x3 = 3 →−7x2 − x3 = −15



2x1 − x2 + 3x3 = 3
3x1 + 2x2 − x3 = 6 −7x2 − 7x3 = −21

 x1 + 3x2 + 2x3 = 9 x1 + 3x2 = 7  x1 = 1

−
−→−7x2 − x3 = −15 −→−7x2 = 14 −→ x2 = 2

 6x3 = 6  x3 = 1  x3 = 1



x1 + 2x2 + 3x3 + 4x4 = −3
x1 + 2x2 + 3x3 + 4x4 = −3
 x1 + 2x2 − 5x4 = 1 −3x3 − 9x4 = 4
→
 2x1 + 4x2 − 3x3 − 19x4 = 6 −9x3 − 29x4 = 12

3x1 + 6x2 − 3x3 − 24x4 = 7 −11x3 − 36x4 = 16



1
x1 + x2 + 3x3 + 4x4 = −3
→
−3x3 − 9x4 = 4

 x1 = 1 − 2t1 + 5t2
x2 = t1 x1 = 1 = 2t1 + 5t2  x2 = t1

→ 4 → 4
( x4 = t2 ( x3 = −− 3t2  x3 = −− 3t2
3  3
 x4 = t2






(1)
i ↔
j

(2)
c ×
i

(3)
c ×
i +
j

℄
Æ

℄ 2