{ 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(, ) = || ||.