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第三章 线性方程组 PowerPoint PPT Presentation


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第三章 线性方程组. §1 消元法 §2 n 维向量空间 §3 线性相关性 §4 矩阵的秩 §5 线性方程组有解判别定理 §6 线性方程组解的结构. §1 消元法. 现在讨论一般线性方程组: (1) 其中 为 n 个未知量, s 为方程个数; 为.

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第三章 线性方程组

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6149132

  • 1

  • 2 n

  • 3

  • 4

  • 5

  • 6


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1

  • (1)

    ns


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sn1


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A


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  • 9-1-6


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1

2

3

1 123


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  • 1

    1

    3 21


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3


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1

4

4


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  • 1r=n


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2r<n


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7


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7 1

  • r>n

  • r=nr<n


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1

s<n

rs<nr<n


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Back


6149132

2 n


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2Pn

Pn

1

1

Pn=2n=3

n>3

3n


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  • 4


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5 (0,0,,0)0

  • 6


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  • 7 kP

    k


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  • 8 PnPn

Back


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3

  • Pn

  • 9 P


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n

n


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  • 10


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  • 1

    2

    3


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11


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(

  • 11` P


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  • 11

    11`


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11`

11


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12


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  • n


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, 3-1-1

2

11`

3


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4

, 4

  • 2n+1

    5


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  • 5

    6

  • 6446


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  • 2

1

2r>s


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1


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r>s

1


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  • 1

    rs.

  • 2n+1n

    n+1n

  • 3 ,


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13


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  • 3

    23


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  • 14

  • 2

Back


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4

  • 15


6149132

3A

3


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1

r<n

  • 1

    A


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rr

1

2

1221||


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  • 4

  • A=r=

    A


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rr

ArA

||


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  • 5

    An

  • AnAnn=1A|A|=|0|=0. n>1A


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|A|=0

n

n=1|A|=0AA n-1n AA

Ann n


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|A|=0n-1


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n||


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  • 5Cramer

  • 16 snAkk kAk


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  • 6 rrr+1

  • ArAr+1Ar+15


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Ar

rArrr

r


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r rr

Ar

Arr+1Ar


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Ar+1Ar+2r

Att r Art rAtr+1t=r

ArArArAr+1Ar


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r


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  • r(A)=r

Back


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5

  • 1


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1

1


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  • 1


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  • 1

    A A

    A r r


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1||

  • A


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  • CramerA rArr rr+1,,s1

    4


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  • r=nCramer41

  • r<n4 5

  • 5cramer


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1

Back


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6

  • (1)


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1

2

  • n=3

  • 17 1


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1

11

2

  • 7 n-rrn-r

  • 1rr


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1 3

  • r=n,r<n

    311


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3n-r

(1,0,,0),(0,1,,0),,(0,0,,1) 4

31n-r

5

5


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454r5

(1)

(6)

1 1

7

176n-r


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  • 9

  • 19

  • 19 1

  • 291


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  • 8 99

    10

    19 19

  • 1 1


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9102 1

9||

  • 89

    9

  • 1


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  • 11

    1111

  • 1A=1 =1A


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  • 2 A=1 =2

  • 3 A=2 =2


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t

13

  • 11

    14

    1113


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15

13151114


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R(I)=R(II)=3,R(III)=4.

4


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R(I)=R(II)=3,

1


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1

R(III)=4

4


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