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# 線性方程式系統

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## 線性方程式系統

(2)



1

2

3

n

1

1



(3)



a x + y =

### (

b x + y −π z =

0 10

2 )

(c x1x2 + x3 + x4 =

1

2 2

d π xx = e

e xy + z =

f

ex

y =

0

3 2

sin )

(g x1 + x2x3 =

+

=

y

h x

(4)



b x

a x

a x

a x

a + + +

### L

+ n n =

3 3 2

2 1

1

1,

1 s

x = x2 = s2, x3 = s3,

,

n

n s

x =

b s

a s

a s

a s

a + + +

+ n n =

3 3 2

2 1

1



(5)



4

2 2

1 + x =

x

2

1 4 2x

x = − x2 = t

2

1 = x =

x

t

t

tR

2

1 4 t

x = −

s

21 s

sR

## }

(6)



n n n

n n

b x

a x

a x

a x

a

b x

a x

a x

a x

a

b x

a x

a x

a x

a

b x

a x

a x

a x

a

= +

+ +

+

= +

+ +

+

= +

+ +

+

= +

+ +

+

### L L L

3 3

3 3

33 2

32 1

31

2 2

3 23 2

22 1

21

1 1

3 13 2

12 1

11

m n

mn m

m

m x a x a x a x b

a + + +

+ =

3 3 2

2 1

1





(7)



(8)



=

− =

+ y x

y x

=

+ =

+

y x

y x

x + y =

=

+ =

+ y x

y x

(9)



== −

y y x

y = −2

− ==

x x

2 ,

1 = −

= y

x

(10)



(3) (2) (1) 2

5 3

9 3

2

==

+ =

+

z z y

z y

x

z = 2

=

+ y

== − +

y y

y = −1

z = 2

− + ==

x x

2 ,

1 ,

1 = − =

= y z

x

(11)





(12)



(3) (2) (1) 17

5 5

2

4 3

9 3

2

= +

− = −

+

− − + =

z y

x

y x

z y

x

(1) + (2) → (2)

(4) 17

5 5

2

5 3

9 3

2

= +

− ++ ==

z y

x

z y

z y

x

(5) 1

5 3

9 3

2

(3) (3)

2) (

(1)

=

− ++ ==

→ +

×

z y

z y

z y

x

(13)

(6) 4

2

5 3

9 3

2

(5) (5)

(4)

==

+ =

+

→ +

z z y

z y

x

9 3

2

) 6 ( (6) 21

= +

×

z y

x

x =

y = −

z =

2

5 3

9 3

2

==

+ =

+

z z y

z y

x

(14)



(3) (2) (1) 1

3 2

2 2

2

1 3

3 2

1

3 2

1

3 2

1

=

+−− −+ == x

x x

x x

x

x x

x

(1) ( 2) (2) (2)

→ +

×

→ +

×

) 5 (

) 4 ( 2

4 5

0 4

5

1 3

(3) (3)

) 1 ( (1)

3 2

3 2

3 2

1

=

− =

− =

+

→ +

×

x x

x x

x x

x

(15)

2 0

0 4

5

1 3

) 5 ( )

5 ( ) 1 ( ) 4 (

3 2

3 2

1

==

− =

+

→ +

×

x x

x x

x

(16)



(3) (2) (1)

1 3

1 3

0

2 1

3 1

3 2

= +

− −− == −

x x

x x

x x

1 (1) 3

) 2 ( )

1 (

=

x x

(3) (2) (1)

1 3

0 1 3

2 1

3 2

3 1

= +

− −− == −

x x

x x

x x

(4)

0 3

3

0 1 3

(3) (3)

(1)

3 2

3 2

3 1

=

− =

− = −

→ +

x x

x x

x x

(17)

0 1 3

3 2

3 1

=

=

x x

x x

t x3 =

3,

2 x

x =

3

1 1 3x

x = − +

,

,

, 1 3

3 2 1

t x

R t

t x

t x

=

=

= t x3 =

(18)





















(19)



×

mn m

m m

n n n

a a

a a

a a

a a

a a

a a

a a

a a

3 2

1

3 33

32 31

2 23

22 21

1 13

12 11

m

n

11

22

nn

### 稱為主對角線

(main diagonal)的元素

n

m = n



ij

×n

(20)



]

2 [

0 0

0 0

×

×

×

− 7 4 2 2

π e

2 3×



(21)



n n

n n

n n

b x

a x

a x

a x

a

b x

a x

a x

a x

a

b x

a x

a x

a x

a

b x

a x

a x

a x

a

= +

+ +

+

= +

+ +

+

= +

+ +

+

= +

+ +

+

### K K K

3 3

3 33 2

32 1

31

2 2

3 23 2

22 1

21

1 1

3 13 2

12 1

11

m n

mn m

m

m x a x a x a x b

a + + +

+ =

3 3 2

2 1

1

= nn

n

a a

a a

a a

a a

a a

a a

a a

a a

A

3 33

32 31

2 23

22 21

1 13

12 11

= bm

b b

b

2 1

= xn

x x

x

2 1

b Ax =

(22)



3

2 1

3 2

1

3 33

32 31

2 23

22 21

1 13

12 11

b A b

b b b

a a

a a

a a

a a

a a

a a

a a

a a

m mn

m m

m

n n n

=



A a

a a

a

a a

a a

a a

a a

a a

a a

mn m

m m

n n n

=

3 2

1

3 33

32 31

2 23

22 21

1 13

12 11

(23)



j i

ij R R

r : ↔

i i

k

i k R R

r( )

j j

i k

ij k R R R

r( ) : ( ) + →

(row equivalent)



(24)



1 4 3 2

4 3 1 0

3 0 2 1

− −

1 4 3 2

3 0 2 1

4 3 1

0 r12

− −

0 3

3 1

1 3

2

1

− − − 0 3

3 1

2 6

4

2 ( )

1

2

r 1

− −

2 1

2 5

0 3

3

1

− −

2 1

2 5

0 3

3 1

− −− − 8 13

3 0

1 2

3 0

3 4

2 1

−−

−−

2 5

1 2

1 2

3 0

3 4

2

1 ( 2)

13

r

(25)



17 5

5 2

4 3

9 3

2

= +

= +

= +

z y

x

y x

z y

x

17 5

5 2

4 0

3 1

9 3

2 1

17 5

5 2

5 3

9 3

2

= +

= +

= +

z y

x

z y

z y

x

1 1

1 0

5 3

1 0

9 3

2 1

2 2

1 )

1 (

12 :(1)R R R

r +

3 3

1 )

2 (

13 :( 2)R R R

r +

1 5 3

9 3

2

=

= +

= +

z y

z y

z y

x

(26)



3 3

2 )

1 (

23 :(1)R R R

r +

=

= +

= +

z z y

z y

x

2 1 1

= −

== z y x

2 1 0

0

5 3

1 0

9 3

2 1

3 3

2) (1

3 )

2 (1

: R R

r

2 5 3

9 3

2

=

= +

= +

z z y

z y

x

(27)





(28)



− − −

2 3

1 0 0

3 1

2 5 1

3 1

0 0

2 0

1 0

1 0

0 1

− 1 0

0 0 0

4 1

0 0 0

2 3

1 0 0

0 0 0 0 3 1

0 0

2 0

1 0

−−

(29)







(30)

4 5

4 1 8

2

12 8

0 2 0

0

28 12

4 6 8

2

r12



4 5

4 1 8

2

28 12

4 6 8

2

12 8

0 2 0

0

4 5

4 1 8

2

12 8

0 2 0

0

14 6

2 3 4

) 1

( 1

2

r 1

24 17

0 5 0

0

12 8

0 2 0

0

14 6

2 3 4

1

) 2 ( 13

r

(31)

24 17

0 5 0

0

6 4

0 1 0

0

14 6

2 3 4

) 1

( 2

2

1

r

6 3

0 0 0

0

6 4

0 1 0

0

14 6

2 3 4

1

) 5 ( 23

r

1 4 3 2 6 14

3 ) ( 1

r

1 4 3 2 0 2

) 6 (

r

2 1

0 0 0

0

6 4

0 1 0

0

14 6

2 3 4

1

3

r3

2 1 0 0 0

0

2 0 0 1 0

0

2 0 2 3 4

1

) 4 (

r32

2 1 0 0 0 0

2 0 0 1 0 0

8 0 2 0 4 1

) 3 (

21

2 1

0 0 0

0

6 4

0 1 0

0

2 0

2 3 4

1

) 6 ( 31

r

(32)



17 5

5 2

4 3

9 3

2

= +

− = −

+

− − + =

z y

x

y x

z y

x

### 

1 − 2 3 9 (1) (2)

1 − 2 3 9

r(1)

1 − 2 3 9

− −

− −

17 5

5 2

4 0

3 1

9 3

2 1

2 1

0 0

1 0

1 0

1 0

0 1

) 9 ( 31 )

3 ( 32 ) 2 (

21 , r , r r

2 1 1

= −

== z y x

1 1

1 0

5 3

1 0

9 3

2 1

) 2 ( 13 ) 1 (

12 ,r

r r23(1)

4 2 0 0

5 3 1 0

9 3 2 1

2 1 0 0

5 3 1 0

9 3 2

) 1

2 (1

r3

(33)



1

5

3

0 2

4 2

2 1

3 1

1 ++ − ==

x x

x x

x

1 0 5

3

0 2 4

2

− 3 1 1

0

2 5

0 1

### 列簡梯形形式

) 2 ( 21 ) 1 ( 2 ) 3 ( 12 ) (

1 21 ,r ,r ,r r

3 5 0 1

0 1 − 3 −1

3 2

3

1 + − == −

x x

x x

3

2 1

x

x x

(34)

3 2

3 1

x x

x x == − +−

x3 = t

2 1

R t

t x

t x

∈ +

=

=

3 t

x =

(35)



2 3

23 2

22 1

21

1 3

13 2

12 1

11

= +

+ +

+ + + + =

+ + + + =

+

n n

n n

x a x

a x

a x

a

x a x

a x

a x

a

x a x

a x

a x

a

3 3 2

2 1

1

3 3

33 2

32 1

31

= +

+ +

+

= +

+ +

+

n mn m

m m

n n

x a x

a x

a x

a

x a x

a x

a x

a

(36)





3

### 0

2

1 = x = x = = xn =

x



(任意n變數齊次系統的解)

(37)



3 2

1

3 2

1 +− ++ ==

x x

x

x x

x

0 3 1 2

0 3 1 1

−1 0 1

0

0 2 0

1

### 列簡梯形形式

) 1 ( 21 ) ( 2 ) 2 (

12

r 13

r r

2 1 3 0

0 1 −1 0

3 2 1

x

x x

x3 = t

R t

t x

t x

t

x1 = −

2 =

3 =

1 2 3

t = x = x = x =

(38)





















(39)























(40)

### 1.3 線性方程式系統的應用

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 Method 2 ：利用兩平面法向量外積可得直 線方程式的方向向量，再找直線上一點，可得

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