.


:




:

































 

 

 

 





 

{ 1, 2, , k } L :

1= ( 11, 21, , 1),

2= ( 12, 22, , 2),

..................

k = ( 1 k, 2 k , , k).

, .. , :

.

, . , .

3

k - , , k.

, . 1:{ 1, 2, , } 2:{ } L. L, 2 1.

3

2 1 1 2 .

. (2.2)

.

L. 1 2:

, .

. :

= = × ,

= × , = × .

 

17

 

: . . . R n.

 

L. .

1

, b Î L , (, b)

1. (, b) = (b, ),

2. ( + , b) = (, b) + (, b),

3. (a , b) = a(, b)

4. > 0 " ¹ 0 = 0 Û = 0,

, (, b) b.

, . . .

1) 4) : (), , , Û .

2

, .

.

1) 4) .

.

V2 V3 , .. , ,

.

R (x)

.

.

1) .

2) . ,

.

3)

.

4) . , , . , , (.. ) Û , .

, , , R (x), .

R n

.

, , .. . L n { 1, 2, , }.

= a1 1+ a2 2+ + a b = b1 1 + b2 2 + + b .

(, b) = a1b1+ a2b2+ + a b . (*)

:

1) (, b) = a1b1+ a2b2+ + a b = b1a1+ b2a2+ +b a = (b, ),

2) ,

.

( + , b) =

= (, b) + (, b).

3. (l , b) = (la1)b1+ (la2)b2+ + (la )b = la1b1 + la2b2 + + la b =

= l(a1b1) + l(a2b2) + + l(a b ) = l (, b).

4. " ¹ 0 , a i = 0, .. = 0.

, (, b) = a1b1 + a2b2 + + a b L n .

, (, b) = a1b1+ a2b2+ + a b b. , - . (*) .

3

.

|| ||, [ ], | |. , ,

|| || .

:

1. || || = 0 Û = 0.

2. ||a ||= |a|.|| || "a ÎR.

3. |(, b)| £ || ||.|| b || ( - ).

4. || + b || £ || || + || b || ( ).

V2 V3 `

||` || = |` |.

R n

|| a || = .

4

( ), : || a || = 1.

¹ 0, . ( ) .

,

, ,

.

5

j (0£ j <p), cosj = , b .

, b

j = = arccos .

, , , , , . , , , .

6

b , :

= 0.

, , . , .. 0 = 0. , (0, b) = (0. , b) = 0.(, b) = 0. , .

7

1, 2, , , , ..

( i, j) = 0 " i ¹ j, i, j =1,2,, m.

1, 2, , ( ), , ..

( i, j) = , i, j = 1,2, , m.

:

1. , , .

2. .

, , , ? .

3

- () .

. .

{ 1, 2, , n }, { g 1, g 2, , g n }, , .. . { 1, 2,, n } .

, :{ 1, 2, , n } .

1.

g 1 = 1, g 2 = 2 + g 1

, g 2 g 1, .. (g 1, g 2) = 0.

,

= .

g 2 = 2 g 1 g 1.

g 3 = 3 + g 1 + g 2,

, g 3 g 2, g 3, .. (g 1, g 3) = 0 (g 2, g 3) = 0.

,

.

.

, g 3 = 3 ` g 1 g 2 g 1 g 2.

g 4 = 4 ` g 1 g 2 g 3.

, (g 1, g 4) = 0, (g 2, g 4) = 0, (g 3, g 4) = 0.

,

g = g 1 g 2 g 1,

, g 1, g 2, , g 1, .

, { g 1, g 2, , g },

g 1 = 1,

g k = k g 1 g 2 g k 1, k = 2, 3, , n,

, , 5.2, , , , - .

2. , , , , . 5.1 5.2, 1, 2, , n , , .

, . ▲

( ):

1) .

2)

g 1 = 1,

g k = k g 1 g 2 g k 1, k = 2, 3, , n.

3) { g 1, g 2, , g }, .. .

4) { 1, 2, , n }.

0:{ 1, 2, , n }.

.

1) : (, b) = a1b1 + a2b2 + + a b .

2) , .

, , .

3) .

|| a || = .

 

8.

, , r(, ) , :

1. r(, ) = r(, );

2. r(, )³0 , r(, )=0 , = ;

3. r(, ) £ r(, z) + r(, z) , , z Î.

.

 

R n, ( ) r(, ) = || ||.





:


: 2016-10-22; !; : 5373 |


:

:

, .
==> ...

809 - | 733 -


© 2015-2024 lektsii.org - -

: 0.103 .