, , , (x, y). n - .
. n R n
x = (x 1 ,..., x n )
x 1,..., x n . R .
. , . , .
, . x
x =
x -.
x = (x 1 ,..., x n) y = (y 1 ,..., y n ) , x 1 = y 1 ,..., x n = y n .
x y R a - ,
x + y = (x 1 + y 1 ,..., x 1 + y n ),
a x = (a x 1 ,..., a x n ),
, x + y a x R .
, x = (3, -1, 2) y = (2, 4, 1) R ,
2 x + 3 y = (6, -2, 4) + (6, 12, 3) = (6 + 6, -2 + 12, 4+3)=(12, 10, 7)
R .
, .
:
1.
x + y = y + x
2.
(x + y) + z = x + (y + z)
3.
(ab) x = a(b x)
4.
(a + b) x = a x + b x
5.
a(x + y) = a x + a y.
6. 0 =(0,...,0)- ,
x + 0 = 0 + x = x.
7.
1× x = x
8. , .. , b x,
a + x = b.
R , , .
. R . .
L , :
) x, y ÎL , x+y;
) x ÎL a a x ÎL, a x;
) 1) - 8).
. n-1 .
|
|
. X R , x ÎX y ÎX , a x + b y X a b.
. X Ì R - . X R . , x = (x , x , 0) y = (y , y , 0) X,
a x + b y = (a x 1+ b y 1 , a x 2+ b y 2 , a0 + b0) = (a x 1+ b y 1 , a x 1 + b y 2 , 0)
X.
R .
. x = (x 1 ,..., x n ) y = (y 1 ,..., y n ) R ( (x, y)),
(x, y) = x 1 y 1 +... + x n y n
, :
1. (x, y) = (y, x);
2. (x + y, z) = (x, z) + (y, z);
3. (a × x, y) = a × (x, y).
a b , (a, b) = 0.
,
e 1= (1, 0) e 2 = (0, 1)
. .
( 1 , 2 ) =1×0 + 0×1 = 0 e 2
e
. 3
a 1,..., a m , (a k, a l ) = 0 k¹ l.
.
, n . i x i , x = (x 1 ,..., x n ) . i - i, c = (c 1 ,..., c n ) .
(c, x) = c 1 x 1 +... + c n x n
. i y i, y = (y 1 ,..., y n ) . y - x .
R.
, ( ). , .
. a 1 ,..., a m , a 1 ,..., a m ,
a 1 a 1+...+ a m a m = 0. (1)
, (1) , a 1 ,..., am .
, a 1,..., a m b 1 ,..., b , . ,
a1 a 1+...+ a m a m = 0
a1,..., a ,
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|
a1 a 1+...+ a m a m +0× b 1 +... + 0× b = 0
a1,..., a , 0,..., 0 .
, 0 , , , . ,
1× 0 + 0 × a 1+... + 0 × a m = 0
a 1,..., a m.
R
e 1 = (1, 0, 0,..., 0, 0)
e 2 = (0, 1, 0,..., 0, 0)
...................
e n - 1= (0, 0, 0,..., 1, 0)
e n = (0, 0, 0,..., 0, 1),
. 1,... n . ,
a 1 1 +... + a n n = (a ,..., a n) = 0 = (0,..., 0),
a1 =... = a n = 0. a 1,..., a . ,
a 1 a 1+...+ a m a m = 0,
0 = (a k , 0)= (a k , a1 a 1+...+ a m a m) = a k (a k , a k ),
( k , k ) ¹ 0 , , a k = 0 k = 1, 2,... m.
b
b = a1 a 1+...+ a m a m
, b a 1,..., a m b a 1,..., a m.
1.1. , , a 1,..., a m , .
. a 1,..., a m . a1,..., a m ,
a1 a 1+...+ a m a m = 0.
a , . k =1,
a1 1 = 0 (a1 ¹ 0),
1 = 0 , . , 1< k £ m,
a1 1 +... + a k k = 0 (a k ¹ 0),
k = - a a 1 1 -... - a a k - 1 k - 1.
, . . ,
k = b 1 1 +... + b k - 1 k - 1,
b 1 1 +... + b k - 1 k - 1 + (-1) k + 0 k + 1+... + 0 m = 0
k .
. a 1 ,..., a m , b , ..
b = a 1 a 1+...+ a m a m
. .
R 1 ,..., n , . , b = (b 1 ,..., b n )
b = b 1 e 1+...+ b n e n ,
.
. 1 ,..., m k , , , k , .
. , b
b = a 1 a 1 +... + a k a k +... + a m a m , (2)
k
k = b 1 1 +... + b k - 1 k - 1 + b k+ 1 k+ 1 +... + b m m . (3)
(3) (2) ,
b = (a1 + a k b 1 ) a k +... + (a k - 1 + a k b k - 1 ) a k - 1 +
+ (a k +1+ a k b k+1) a k +1 +... + (a m + a k b m ) a m .
|
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b , , k , .
1.2. .
. , , , . , 1.1 , , .
1.3. R .
. ,
a 1,..., a m (4)
b 1 ,..., b l , (5)
. , , l > m.
b 1 , a 1 ,..., a m . (6)
(4) , (6) . (6) , b 1 . (6) 1.1, , , k , . (6) ,
b , a ,..., a (7)
,..., (4). (7) .
b 2 , b 1 , a ,..., a . (8)
, b 2 . , 1.1 . ,..., , b 1 b 2 .
b 2 , b 1 , a ,..., a .
, m
b m ,..., b 1 . (9)
, , b m +1, (9), (5). .
1. R n .
, 1 ,..., n .
2. R , n , .
, n , , , , n .
3. , n , R .
, 1 , ..., n . b 1 ,..., b n. a 1 ,..., a n , b 1 ,..., b n . , , , . . . n , b 1 ,..., b n , . . k . 1 ,..., n .
|
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. X R . 1.3, , . , X, .
1 ,..., n - . . , . ,
= a 1 1 +... + a n n
= b 1 1 +... + b n n .
,
0 = (a1 - b 1) 1 +... + (a n - b n) n
- , a 1 - b 1 = 0,..., a n - b n = 0, a 1 = b 1 ,..., a n = b n . a 1,..., a n 1 ,..., n . 3.
2
.
, , m n , (m, n). , . . (m, n)
= , = .
- , i j. : || a ij || mn .
, - , , - , , . , , .
, , .
, .
, a a . ,
7× = ×7 = .
, , , , . ,
+ = .
:
1. (ab) = a(b);
2. A +(B + C) = (A + B) + C;
3. A + B = B + A;
4. (a + b)A = aA + bA;
5. a(A + B) = aA + aB.
(-1) = -. + (-) - .
. . , . - (m, k)-, - (k, n)-
= , = ,
(m, n)- = || ij || m n ,
cij= ai1 b 1j + ai2 b 2j +... + ain bnj (i = 1,..., m; j = 1,..., n).
,
= = .
, , . ,
= ,
= .
, , , - .
, . . ,
= = ,
[ 3, 5, 1] × =
= [ 3´1+5´2+1´3, 3´3+5´1+1´4, 3´1+5´2+1´2] == [ 16, 18, 15].
.
6. a() = (a); (a) = (a); ()a = (a).
7. ( + ) = + .
8. ( + ) = + .
9. () = ().
.
=
(m, n).
=
(n, m), , . .
|
|
.
= .
=
(a + b) = a + b
() = .
= ,
= - ,
. , , (i j = j i ), (i j = - j j).
.
=
,
=
.
.
, . , n. .
, 1, - , .
= .
,
= = ,
. , - n ( - -)
= ,
= .
, ,
= = .
, , , , . , , , . , Y,
X = XE= X(AY) = (XA)Y = EY = Y
, X = Y.
, , - 1. ,
- 1= - 1 = .
. (a) , , i - , . . i i, a. ii (a) ii (a), i. i a ij (j = 1,..., n), . . i a. ii (a) .
.
= , 22(3) = .
22 (3) = = .
i j (a) (i ¹ j) , i j. , , i j (a) a. ij (a) i. i k + a j k (k = 1,..., n), . . i j, a. ij (a) .
. , , 23 (2).
23 (2) = = = .
, . , , .
,
(a ) × × × (a ).
,
(a ) × × × (a ) = ,
,
( (a ) × × × (a ))× = .
,
(a ) × × × (a ) = .
.
.
= .
,
.
, . 1, 0,0. , , .
. ( 1/2).
.
, 2, , 3.
.
, , 2.
.
.
= .
.
=
= .
R .
, , .
n n, .. .
2.1. , .
. . i - .
a +... + a = 0
(a ,..., a ) = 0.
, , , (a ,..., a ) = 0, . . a =... = a = 0. - .
, . R . , . . a i j (i,j=1,,n) ,
a 11 1 +... + a 1 n = e 1
a n1 1 +... + a n n = e n