vector spaces
DESCRIPTION
Vector SpacesTRANSCRIPT
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Version 2.0
2002
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0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 4
1.1 . . . . . . . . . . . . . . . 4
1.2 . . . . . . . . . . . . . . 7
1.3 . . . . . . 8
1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 . . . . . . . 11
1.6 . 15
1.7 . . . . . . . . . . . 19
1.8 . . . . . . . . . . . . . 23
1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 32
2.1 . . . . . . . . . . . . . . . . . 32
2.2 . . . . . . . . . . . . . . . . 36
2.3 . . . . . . . . . . . . . 40
2.3.1 Gram-Schmidt . . . . . . . . . . . . . . . . . 40
2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1
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0.1
:
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, -
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. ,
-
,
, . -
( ) ,
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. -
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. ,
, -
,
,
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0.2
. (
)
:
, ,
, ,
, ,
, -
. (-
).
-
(
).
2
-
:
0.3
.
// / : -/-/- / : ( ) -
. p - q p q p, q q, p.
N R C K R, C .
3
-
1
1.1
1.1.1 K ( R, C) U ( -) :
) (. ), (-) U U U , (. : U U U)) (. ) K U U , (. : K U U). x,y, z U , K:A1 1,
(x y) z = x (y z) x,y, z U
A2 ,
x y = y x x,y U
A3 ,
e U : x e = e x = x x U
( , 2)
0, x 0 = x x U1
X , a, b X, a b X
4
-
A4 x x U ,
x U , x U : x x =
-
x,
x (x) = 0 x U
A5 , ,
(x y) = x y x,y U K2
A6 , ,
(+ )x = x x x U , K
A7 -,
()x = (x) x U , K
A8 -,
1x = x x U
1.1.1 R, - C, .
1.1.2 -
(. . U , V ), K (. . , ,) (.. x, y, a) .
1.1.3
+,
.
2
, x x.
5
-
1.1.4 2 ( -
) , .
,
, .
A5, x,y U
(1 + 1)(x y) = (1 + 1)x (1 + 1)y A6= (1x 1x) (1y + 1y) A8= (x x) (y y) (1.1) AA6
(1 + 1)(x y) = 1(x x) 1(y y) A5= (1x 1y) (1x 1y) A8= (x y) (x y) (1.2) (1.1) (1.2)
(x x) (y y) = (x y) (x y) (1.3) x (1.3)
x [(x x) (y y)] = x [(x y) (x y)] A1[x (x x)] (y y)] = x [(x y) (x y)] A1[(x x) x] (y y) = [(x x) y] (x y)] A4
(0 x) (y y)] = [0 y] (x y)] A3x (y y) = y (x y) A3 (1.4), y (1.4)
[x (y y)] (y) = [y (x y)] (y) A1x [(y y) (y)] = y [(x y) (y)] A1x [y (y (y))] = y [x (y (y))] A4
x (y 0) = y (x 0) A3x y = y x
1.1.5 ,
1 2
.
6
-
1.2
R2 =R R = {a = (a1, a2) : a1, a2 R} ,
a b = (a1, a2) (b1, b2) = (a1 + b1, a2 + b2) a,b R2
+ ,
a = (a1, a2) = (a1, a2) a R2 R a = (a1, a2), b = (b1, b2), c = (c1, c2) R2 , R A1 ():(a b) c = [(a1, a2) (b1, b2)] c) = [(a1 + b1, a2 + b2)] (c1, c2) =
= ((a1 + b1) + c1, (a2 + b2) + c2) == (a1 + (b1 + c1), a2 + (b2 + c2)) = (a1, a2) (b1 + c1, b2 + c2) == a [(b1, b2) (c1, c2)] = a (b c) A2 . A3 ( ) (0, 0) R2
a (0, 0) = (a1, a2) (0, 0) = (a1 + 0, a2 + 0) = (a1, a2) = a a R2
0 = (0, 0).A4 ( ): , a = (a1, a2), (a1,a2) ,
(a1, a2) (a1,a2) = (a1 + (a1), a2 + (a2)) = (0, 0) = a R2
A5 ( ): R a,b R2
(a b) = [(a1, a2) (b1, b2)]) = (a1 + b1, a2 + b2) == ((a1 + b1), (a2 + b2)) = (a1 + b1, a2 + b2) == (a1, a2) (b1, b2) = (a1, a2) (b1, b2) == a b
A6 ( ): , R a R2
(+ )a = (+ )(a1, a2) = ((+ )a1, (+ )a2) == (a1 + a1, a2 + a2) = (a1, a2) (a1, a2) == (a1, a2) (a1, a2) = a a
A7 ( ): , R a R2
()a = ()(a1, a2) = (()a1, ()a2) = ((a1), (a2)) == (a1, a2) = ((a1, a2)) = (a)
7
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A8 ( -): , a R2
1a = 1(a1, a2) = (1 a1, 1 a2) = (a1, a2) = a
R2 .
Rn, nm -, F(R) R R( F(R) = {f(x) : R R R 3 x 7 f(x) R}) , A , .
. , , -
, -
, .
R2 -
a = (a1, a2) = (a1, 0), a R2 R, 8:
1a = 1(a1, a2) = (a1, 0) 6= a
-
.
1.3
8
, :
1.
x a = x b a = b a,b,x U
x a = x b x+ (x a) = x (x b) (x x) a = (x x) b 0 a = 0 b a = b
8
-
1.3.1
. .
2. 0
0 = 0 K
0 = 0 0 (1.5)
0 = (0 0) = 0 0 (1.6) (1.5) (1.6)
0 0 = 0 0
.
3. 0
0a = 0 a U
0a = (0 + 0)a = 0a 0a (1.7)
0a = 0 0a (1.8) (1.7) (1.8)
0a 0a = 0 0a
.
4. 1
(1)a = (a) a U
(1)a a = (1)a 1a = (1 + 1)a = 0a = 0
9
-
5.
a = 0 = 0 a = 0 6= 0,
x = 0 1(x) =
10
(1
)x = 0 x = 0
x 6= 0, 6= 0
x = 0 1(x) =
10
(1
)x = 0 x = 0
, = 0
6.
(a b) = (a) (b)
7.
(a) = a
8.
a = b 6= 0 a = b
9.
a = a a 6= 0 =
.
1.4
1.4.1 - V U U , - U . , :
a,b V , a b V K a V , a V .
1.4.1 , V , U U .
10
-
V -, , a. 0a = 0.
1.4.1 (1.4.1)
(1.4.1) :
{0} .
1.4.2 {0} U .
1.4.3 U - U .
1.4.4 U - -,
U .
,
1.4.2 - V U -
, K a,b V a b V
: V U , .
: = = 1 , = 0 .
1.5 -
1.5.1 R2 V = {a =(x, y)/a R, x + 3y = 0}, : , R2 x+ 3y = 0. V - -, 0 = (0, 0), R2. , 0 + 3 0 = 0. ,a = (x1, y1), b = (x2, y2) V . , , R
a b = (x1, y1) (x2, y2) = (x1, y1) (x2, y2) == (x1 + x2, y1 + y2) (1.9)
11
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(x1 + x2, y1 + y2)
x1 + x2 + 3(y1 + y2) = x1 + 3y1 + x2 + 3y2 =
(x1 + 3y1) + (x2 + 3y2)a,bV= 0 + 0 = 0
: R2 {a = (x, y)/a R, x+ y = 0} R2 Rn
{x = (x1, x2, . . . , xn)/x Rn, 1x1 + 2x2 + + nxn = 0} Rn ( -) Rn , .
3
,
.
1.5.2
V = {X/X,M Rnn,XM = O}( O n n) , nn , M , O. V Rnn. V - - Rnn. , OM = O. , A, B V . , R A,B V (A+ B)M = (A)M+ (B)M = (AM) + (BM) = O+ O = O4
, XM =O , -
( M = AB, mij = ai1b1j + ai2b2j + + ainbnj).3
4
+ . , (. .
, )
+ , .
12
-
.
,
. R2
V1 = {a = (x, y)/a R2, 4x+ 3y = 1}V2 = {a = (x, y)/a R2, |x|+ |y| > 1}V3 = {a = (x, y)/a R2, x 2y}V4 = {a = (x, y)/a R2, y = x2}
R2. V1, V2 - , 0 = (0, 0) R2, V3, V4 (. . v = (5, 3) V3, 5 2 3 = 6, 2v = (10, 6) / V3, 10 2 (6) = 12 u = (1, 1) V4, 1 = 12, 3u = (3, 3) / V4, 3 6= 32). ,
V1 ( , ).
,
T , - , F(R) ( ).
,
1.5.1 V1, V2, . . . , Vn U U . , V =
ni=1 Vi U .
V -, - U ( U , U ). a,b V = ni=1 Vi. , R V1, V2, . . . , Vn , a b V1, V2, . . . , Vn .
1.5.1 -
. Vx ={a = (x, 0)/x R} Vy = {a = (0, y)/y R} R2, , v1 = (1, 0) V1 V2 v2 = (0, 1) V1 V2, v1 + v2 = (1, 1) / V1 V2.
(1.5.1) to
13
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1.5.3
m n Rn.
m n : a11x1 + a12x2+ +a1nxn = 0 am1x1 + am2x2+ +amnxn = 0 m Rn, (1.5.1).
,
m Rn, Rn.
1.5.2
, {0} . U - , S = {v1,v2, . . . ,vn} S U . , K vi S i = 1, 2, . . . . vi S . vi, 2vi, 3vi, . . . , (n + 1)vi , S. S n , .
,
1.5.3 U K , V W . V W = {v w/v V,w W} U V W .
0 V W ( U) 0 0 = 0 V W , V W -. x1,x2 V W . x1,x2 V W , v1,v2 V w1,w2 W , , v1 w1 = x1 v2 w2 = x2. , K x1 x2 = (v1 w1) (v2 w2) = (v1 w1) (v2 w2) =
= (v1 v2) (w1 w2) V , W , v1 v2 V w1 w2 W ,
x1 x2 = (v1 v2) (w1 w2) V W
14
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Y U , V , W V W , V W , u, Y (V,W Y &V W U). u V W , v V w W , , u = v w. v V w W V , W Y , v,w Y Y , v w = u Y , . V W U V , W .
1.5.2 V W V W . 1.5.3 (1.5.3)
( ) .
1.6 -
1.6.1 v U , - K, v1,v2, . . . ,vn 1, 2, . . . n K,
v = 1v1 2v2 nvn 1.6.1 a = (5,3) R2 - a1 = (0, 2), a2 = (1, 2).,
72 (0, 2) + (5)(1, 2) = 72a1 + (5)a2
1.6.2 v = (9, 2, 7) R3 v1 = (1, 2,1), v2 = (6, 4, 2) R3. 1, 2 K,
v = (9, 2, 7) = 1v1 + 2v2 = 1(1, 2,1) + 2(6, 4, 2) == (1 + 62, 21 + 42,1 + 22) (1.10), 1 + 62 = 921 + 42 = 21 + 22 = 7 1 = 3 2 = 2.
15
-
1.6.2 v1,v2, . . . ,vn U . v U v1,v2, . . . ,vn, v1,v2, . . . ,vn U .
1.6.3 v1 = (1, 0), v2 = (1, 1) R2 R2. v = (x, y) R2 - v1 v2, , 1, 2 K,
v = (x, y) = 1v1 + 2v2 = 1(1, 0) + 2(1, 1) = (1, 1 + 2)
{1 = x
1 + 2 = y{
1 = x2 = y x v1,v2 R2
1.6.3 v1,v2, . . . ,vn U . v1,v2, . . . ,vn 1, 2, . . . , n K,
1v1 2v2 nvn = 0
1.6.4 v1,v2, . . . ,vn U . v1,v2, . . . ,vn
1v1 2v2 nvn = 0 1 = 2 = = n = 0
1.6.4 v1 = (1, 1, 1), v2 =(1, 1, 0), v3 = (1, 0, 0) .
1, 2, 3 K ,
1v1 + 2v2 + 3v3 = 0 (1 + 2 + 3, 1 + 2, 1) = (0, 0, 0)
, 1 + 2 + 3 = 01 + 2 = 01 = 0
1 = 2 = 3 = 0
.
16
-
1.6.5 v1 = (1, 1),v2 = (2,2) .
2v1 + v2 = 0 .
1.6.1 (1.6.4), (1.6.5)
m Rn, x111 + x122+ +x1nn = 0 xm11 + xm22+ +xmnn = 0 .
,
, . m > n, .
n ( ) n .
1.6.6 f1(x) = x f2(x) = cosx F(R), R R .
1, 2 R, ,
1x+ 2 cosx = 0 x R (1.11) x1, x2 R, (1.11) 1 = 2 = 0, () . , x1 = pi3 x2 =
pi6 ,
1pi
6+ 2 cos
pi
3= 0 pi1 + 322 = 0
1pi
6+ 2 cos
pi
6= 0 pi1 +
32 = 0
1 = 2 = 0, x cosx .
1.6.2 (1.6.6)
2 ( n) f1(x), f2(x)( f1(x), f2(x), . . . , fn(x)) 2 ( n) x1, x2 ( x1, x2, . . . , xn), 2 2( n n) aij = fi(xj), .
17
-
1.6.1 v1,v2, . . . ,vn U -
.
: v1,v2, . . . ,vn . 1, 2, . . . , n K
1v1 2v2 nvn = 0 1 6= 0.
v1 =21v2 n
1vn
: v1,v2, . . . ,vn . ,
v1.
v1 = 2v2 nvn (1)v1 2v2 nvn = 0
- -
v1 = (1, 1),v2 = (2,2) (1.6.5) , v2 = 2v1 1.6.7 f(x) = sin2 x, g(x) =cos2 x h(x) = cos 2x .
, cos 2x =cos2 x sin2 x, h(x) = 1g(x) 1f(x).
1.6.2 v1,v2, . . . ,vn U , .
, k - (k < n)
1v1 2v2 kvk = 01v1 2v2 kvk 0k+1vk+1 0nvn = 0 1, 2, . . . , k ,
. v1,v2, . . . ,vn .
18
-
1.7
1.7.1 U v1,v2, . . . ,vn U . v1,v2, . . . ,vn
U . 1.7.1 e1 = (1, 0, . . . , 0), e2 =(0, 1, . . . , 0), . . . , en = (0, 0, . . . , 1) Kn.
e1, e2, . . . , en ,
1e1+2e2+ +nen = 0 (1, 2, . . . , n) = 0 1 = 2 = = n = 0, Kn, x = (x1, x2, . . . , xn) Kn
x = (x1, x2, . . . , xn) = x1e1 + x2e2 + + xnen
1.7.2
E1 =[1 00 0
], E2 =
[0 10 0
], E3 =
[0 01 0
], E4 =
[0 00 1
] K22
E1,E2,E3,E4
1E1 + 2E2 + 3E3 + 4E4 = O[1 23 4
]=[0 00 0
]
1 = 2 = 3 = 4 = 0
K22,
X =[
] , , , K
X =[
]= E1 + E2 + E3 + E4
.
19
-
1.7.1 v1,v2, . . . ,vn U , k > n U .
k x1,x2, . . . ,xk. v1,v2, . . . ,vn U , x1,x2, . . . ,xk v1,v2, . . . ,vn x1 = a11v1 a12v2 a1nvn xk = ak1v1 ak2v2 aknvn (1.12) x1,x2, . . . ,xk 1, 2, . . . , k K ,
1x1 2x2 kxk = 0 (1.12)
1(a11v1 a12v2 a1nvn) k(ak1v1 ak2v2 aknvn) = 0(a111+a212+ +ak1k)v1 (a1n1+a2n2+ +aknk)vn = 0 (1.13) v1,v2, . . . ,vn , (1.13) - , (1.13)
a111 + a212 + + ak1k = 0 a1n1 + a2n2 + + aknk = 0 1, 2, . . . , k. -
1, 2, . . . , k x1,x2, . . . ,xk .
1.7.1 v1,v2, . . . ,vn U , u1,u2, . . . ,uk , k n
1.7.2 v1,v2, . . . ,vn u1,u2, . . . ,uk U , k = n.
v1,v2, . . . ,vn u1,u2, . . . ,un ( ) , (1.7.1), k n. u1,u2, . . . ,un v1,v2, . . . ,vn , (1.7.1), n k. = .
1.7.2 U (. dimU). 0, dim{0} = 0.
20
-
1.7.2 (1.7.1), (1.7.2)
(1.7.2), dimRn = n dimK22 = 4. , dimKnm = nm
1.7.3 U dimU = n, n .
1.7.3 U dimU = n, n U .
v1,v2, . . . ,vn U v . (1.7.3) v, v1,v2, . . . ,vn , , 1, 2, . . . , n K ,
v 1v1 2v2 nvn = 0 (1.14) (1.14) 6= 0, = 0, (1.14)
1v1 2v2 nvn = 0 (1.15) v1,v2, . . . ,vn 1 = 2 = = n = 0, . (1.14)
v =(1
)v1
(2
)v2
(n
)vn
, U n , n U .
,
1.7.4 v1,v2, . . . ,vn U , U .
v1,v2, . . . ,vn U , .
v = 1v1 2v2 nvnv = 1v1 2v2 nvn
(1 1)v1 (2 2)v2 (n )vn = 0 v1,v2, . . . ,vn
1 1 = 2 2 = = n n = 0
21
-
1.7.1 1, 2, . . . , n v {v1,v2, . . . ,vn} 1.7.3 A - .
A {0} = (0, 0, 0, . . . ). {a1} = (1, 0, 0, . . . ), {a2} = (0, 1, 0, . . . ), . . . , {aN} = (0, 0, . . . , 0, 1, 0, . . . ) {ai} ,
1{a1}+ 2{a2}+ + N{aN} = {0} (1, 2, . . . , N , 0, 0) = {0} 1 = 2 = = N = 0 -
N . o A .
1.7.4 fn(x) = sinnx, n =1, 2, . . . F(R) .
n = 1 f1(x) = sinx - . f1(x), f2(x), . . . , fk(x) , ,
1 sinx+ 2 sin 2x+ + k sin kx = 0 1 = = n = 0 (1.16) k + 1 f1(x), f2(x), . . . , fk+1(x)
1f1(x) + 2f2(x) + + k+1fk+1(x) = 0 (1.17a)
1 sinx+ 2 sin 2x+ + k+1 sin(k + 1)x = 0d2
dx2 (1.17b)1 sinx 222 sin 2x (k + 1)2k+1 sin(k + 1)x = 0 (1.17c) (1.17b) me (k + 1)2 (1.17c)
((k+1)21)1 sinx+((k+1)222)2 sin 2x+ +((k+1)2k2)k sin kx = 0 , sinx, . . . , sinnx -
((k + 1)2 1)1 = = ((k + 1)2 k2)1 = 0 1 = ((k + 1)2 k)N = 0 (1.17b) k+1 sin(k + 1)x = 0 k + 1 .
22
-
1.7.2 1, . . . , k (1.16) 1, . . . , k (1.17a) (?)
1.7.3 1, cosx, cos 2x, . . . .
1.7.4 sinx, sin 2x, . . . 1, cosx, cos 2x, . . . - , -
,
Fourier -
.
(1.7.4), ,
( )
.
1.7.5
R3,V =
{(x, y, z)/x =
y
2=
z
3
} y = 2x z = 3x, V
(x, 2x, 3x) = x(1, 2, 3) x R. v0 = (1, 2, 3) V , dimV = 1. : R3 , V ( ).
1.8
1.8.1 -
v1,v2, . . . ,vn U , U .
V = {v U/v = 1v1 nvn 1, . . . n K} V - U (0 =0v1 0vn). v = 1v1 nvn u = 1v1 nvn. , K
v u = (1v1 nvn) (1v1 nvn) == (1 + 1)v1 (n + n)vn V v1,v2, . . . ,vn.
23
-
1.8.1 V , U v1,v2, . . . ,vn
span{v1,v2, . . . ,vn} ]v1,v2, . . . ,vn[ 1.8.1 U 3 v1,v2, . . . ,vn T o . v1,v2, . . . ,vn T , ]v1,v2, . . . ,vn[ T .
.
1.8.2 v1,v2, . . . ,vn U , .
1.8.3 v1,v2,u1,u2 U , v1 = u1 u2 v2 = u1 u2 6= 0,
]v1,v2[ = ]u1,u2[
,
: v1,v2, u1,u2 ]u1,u2[ (1.8.1)
]v1,v2[ ]u1,u2[, 6= 0 {
u1 u2 = v1u1 u2 = v2 u1,u2
u1 =
v1
v2 u2 =
v1
v2
]u1,u2[ ]v1,v2[
1.8.4 v1,v2, . . . ,vn U :
1. v1,v2, . . . ,vn -,
]v1, . . . ,vi, . . . ,vn[ = ]v1, . . . , vi, . . . ,vn[ 6= 0 i = 1, 2, . . . , n
24
-
2. v1,v2, . . . ,vn - ,
]v1, . . . ,vi, . . . ,vj , . . . ,vn[ = ]v1, . . . ,vi vj , . . . ,vj , . . . ,vn[
K i, j
3. v1,v2, . . . ,vn
]v1, . . . ,vj , . . . ,vi, . . . ,vn[ = ]v1, . . . ,vj , . . . ,vi, . . . ,vn[ i, j
1. (1.8.2) v1 = vi, v1 = vj = = = 0.
2. (1.8.2) v1 = vi, v1 = vj , = = 1 = 0.
3. -
.
1.8.2 .
1.8.1 V R4,
x1 = (1, 2, 5,1), x2 = (3, 6, 5,6), x3 = (2, 4, 0,2)
x2 = x2 (3)x3 = (0, 0,10,3)x3 = x3 (2)x1 = (0, 0,10, 0)x3 = x
3 (x2) = (0, 0, 0, 3)
x1,x2,x3 , V . dimV = 3.
1.8.5 ( Steinitz) x1, . . . ,xk n- U (k n), xk+1, . . . ,xn,
{x1, . . . ,xk,xk+1, . . . xn}
U .
25
-
xk+1 ]x1, . . . ,xk[, xk+2 ]x1, . . . ,xk,xk+1[ ...
1.8.2 U dimU = n V , dimV dimU .
1.8.3 U dimU = n V dimV = n, V = U .
1.9
1. Rn = {x = (x1, . . . , xn)/x1, . . . , xn R} -
, R2 - .
2. Rnm Cnm n m ,
-
.
3. C2 = {z = (z1, z2)/z1, z2 C} - -
R2 - . Cn = {z =(z1, . . . , zn)/z1, . . . , zn C} C2 .
4. F(R) - R R -
, .
5. A -
,
.
6. F(R)nm n m F(R) -
. Unm
26
-
nm U K,
K.
7. (6)-(9) 1.3.
8. Rn
{x = (x1, . . . , xn)/x R, 1x1 + + nxn = 0}
Rn.
9.
Rnn:. V = {X/X Rnn,X2 = I} []. V = {X/X Rnn,X2 = X} []. (
) []
. (
) []
. (,
,
). []
10. R22:. 2 2 []. - 2 2 []. 2 2 , V = {X/X,A R22,AX =XA} []. V = {X/X,A R22,A2 = I,XA = X} []. V = {X/X,A R22,A2 = I,XA = X} []11. -
:
27
-
. V = {f/f F(R), f(1) = f(1)}, F(R) []. ,
T , F(R) []. R[x] , , F(R) []. Rn[x] = {f R[x]/deg f n},
n, R[x] []. C[z] , - f : C C, F(C) []. Cn[x] = {f C[x]/deg f n} , -
n, C[x] []
12. -
:
. (,
{an} |an| M , n), A []. , A []. V = {a = (x, y, z)/a R3, x 0, z 0}, - R3 []. V = {a = (x, y, z)/a R3, xyz = 0}, R3 []
13. :
. f(x) = 2x, g(x) = 3x, h(x) = 4x []. f(x) = 2x+1 3x, g(x) = 2x + 3x2 []. f(x) = x, g(x) = cosx, h(x) = sinx []. f(x) = |x |, g(x) = |x |, h(x) = |x | []. f(x) = sinx cosx, g(x) = 3 sinx+ cosx []. f(x) = cosx+ (2a+ 1) cos 4x, g(x) = (1 a) cosx+ cos 4x []14. nn A,B,C , n n P, P1AP, P1BP, P1CP .
15. 2 2 A, I,A,A2 A5,A6,A7 .
16. a,b U , (2+)a+(1+2)b 6= 0, R.
28
-
17. n m A1, . . . ,Ak B 6= O, AiB = O, i = 1, 2, . . . , k. A1, . . . ,Ak Rnm.
18. F(R) R[x] -.
: - F(R), fi(x) =xi .
19. fn(x) = enx, n N F(R) .
20. fn(x) = cosnx, n N F(R) .
21. fn(x) = sinn x, n N F(R) . fn(x) =cosn x, n N.22. < < ,
{(x+)2, (x+)2, (x+)2} {(x)(x+), (x)(x), (x)(x)}
23. R[x]:
.
. -
(
).
. ()
Rn[x].: R3[x].
24. Rn, V = {v =(x1, . . . , xn)/v Rn, a1x1 + + anxn = 0, |a1|+ + |an| 6= 0}.25. R22:
. V = {A/AM = O}, M =[1 22 4
]. V = {A/AM =MA}, M =
[0 11 2
]. V =
[
]
29
-
26.
C2. C2, . U {v1, . . . ,vn}, , {v1, . . . ,vn jv1, . . . , jvn} R C.
27. V =](1, 0, 0), (0, 1, 1)[ W =](2, 1, 0), (1, 0, 1)[( R3), V W . .28. V =] cos2 x, sin2x[ ( F(R)), /- cosx, 1,
x2 V ?
29. V =] cosx, sinx[ ( F(R)), cos(x +a), sin(x + a) V . {cos(x +a), sin(x+ a)} V30. sin(x+), sin(x+), sin(x+ , , R .
{cos(x+ a), sin(x+ a)} V31. R3
1.10
-
:
. ,
,
. -
,
1 . - Rn, Cn, , ,
,
R[x], C[x], - n, Rn[x], Cn[x] .
,
, -
- . (
)
30
-
.
-
, .
-
.
, -
.
-
.
(
dimRn = n, dimCn = n 2n , dimRn[x] = n + 1, dimRRnm = nm) ( F(R), F(C), A, R[x], C[x] ).
,
.
31
-
2
2.1
2.1.1 x,y - E (. x,y), x,y x,y, z E R:A1 ,
x y, z = x,y+ x, z xy = x,yA2
xy = y,xA3 x,x 0 - x = 0 2.1.2 E, , .
2.1.1 E .
A2
x,y z = y z,x A1= y,x+ z,x A2= x,y+ x, z A2
x, y = y,x A3= y,x A1= x,y
32
-
2.1.3 x (. x) -
x,x, x =
x,x
2.1.1 E
x = ||x
x2 = x, x = x, x = 2 x,x = 2x2 x = ||x
2.1.2 E
x y2 = x2 + 2 x,y+ y2 (2.1)
x y2 = x y,x y = x,x y+ y,x y == x,x+ x,y+ y,x+ y,y = x2 + x,y+ x,y+ y2 == x2 + 2 x,y+ y2
2.1.2
0,
x,0 = 0 x E
(2.1)
x2 = x 02 = x2 + 2 x,0+ 02 = x2 + 2 x,0 x2 = |x2 + 2 x,0 x0 = 0
2.1.3 ( Cauchy-Schwartz)
E, Cauchy-Schwartz:
| x,y | xy x,y E (2.2)
- x,y .
33
-
x,y , . x,y 6= 0, an - ()
() = x y = x2 + 2 x,y+ 2y2
() - , ,
4 x,y 4x2y2 0 .
x,y , , x = ay, a R. | x,y | = | ay,y | = |a|| y,y | = |a|y2 = |a|yy = ayy = xy, | x,y | = xy, x,y , . x,y 6= 0, x,y = xy x,y = xy.
xx ,
yy= 1
xx ,
yy
+
xx ,
yy
= 1 + 1 =
x2x2 +
y2y2 =
=
xx ,
xx
+yy ,
yy
xx ,
xx
xx ,
yy
+yy ,
yy
xx ,
yy
= 0 (2.3)
xx ,
xx
xx ,
yy
+yy ,
yy
xx ,
yy
=
=
xx ,
xx
+
xx ,
yy
+yy ,
yy
+
xx ,
yy
=
=
xx ,
xx
yy
+
xx
yy ,
yy
=
=
xx
yy ,
xx
+
xx
yy ,
yy
=
=
xx
yy ,
xx
yy
(2.3)= 0
xx yy = 0 x = xyy, x,y . x,y = xy, x = xyy .
34
-
2.1.3 - x,y, 1 x,yxy 1 ,
cos =xyxy 2.1.1
x,y.
2.1.4 x,y , x,y = 0
pi2 (. xy).
2.1.5 x1, . . . ,xn - , ,
xi,xj = 0 1 i 6= j n
2.1.6 x1, . . . ,xn -
xi = 1 1 i n
.
2.1.4
.
x1, . . . ,xn 1, . . . , n 1x1 nxn = 0.
1x1 nxn,xi = 0,xi i = 1, 2, . . . , n1x1,xi+ + nxn,xi = 0 i = 1, 2, . . . , n
1 x1,xi+ + n xn,xi i = 1, 2, . . . , n (2.4)
xi,xj = 0 j 6= i xi,xi = 1. (2.4)
i = 0 i = 1, 2, . . . , n
2.1.7 (.
) -
A E
A = {x E/ x,v = 0,v A}
2.1.5 A E, A E.
35
-
A -, E ( 0,v = 0 v A). x1,x2 A 1, 2 R. v A:
1x1 2x2,v = 1x1,v+ 2x2,v = 1 x1,v+ 2 x2,v
x1,x2 A x1,v = 0 = x2,v.
1x1 2x2,v = 1 x1,v+ 2 x2,v = 0
1x1 2x2 A, , A E.
2.1.4 E , ,
E = {0}
.
2.1.6 ( Minkowski) E Minkowski ( ):
x y x+ y
(2.1)
x y2 = x2 + y2 + 2 x,y (2.5)
x,y | x,y | xy , (2.5)
x y2 x2 + y2 + 2xy = (|x+ y)2 |x y |x+ y
2.2
2.2.1 R3
x,y = x1y1 + x2y2 + x3y3 x = (x1, x2, x3),y = (y1, y2, y3) R3
.
Cauchy-Schwartz. x = (1, 1, 0),y = (1, 0, 1) .
36
-
x,y, z R3 R. ,A1 ( ):
x+ y, z = (x1 + y1)z1 + (x2 + y2)z2 + (x3 + y3)z3 == x1z1 + y1z1 + x2z2 + y2z2 + x3z3 + y3z3 == (x1z1 + x2z2 + x3z3) + (y1z1 + y2z2 + y3z3) == x, z+ y, z
x,y = x1y1 + x2y2 + x3y3 = (x1y1 + x2y2 + x3y3) = x,yA2 ():
x,y = x1y1 + x2y2 + x3y3 = y1x1 + y2x2 + y3x3 = y,xA3: x,x = x21+x22+x23 0 - x1 = x2 = x3 = 0, x = 0. Cauchy-Schwartz
| x,y | xy |x1y1 + x2y2 + x3y3| x21 + x
22 + x
23
y21 + y
22 + y
23
,
x =12 + 12 + 02 =
2, y =
12 + 02 + 12 =
2
cos =x,yxy =
1 1 + 1 0 + 0 1+22
=12 =
pi
3
2.2.1 Rn - x,y =nk=1 xiyi ( ) 2.2.2 Rnn - A,B = tr(ATB), AT A tr() ( ) .
Cauchy-Schwartz.
A,B,C Rnn R. ,A1 ( ):
A+B,C = tr((AT +BT )C) A,C aij , cij , pij P = AC pij =
nk=1 aikckj P = A
TC, pij =n
k=1 akickj ( AT
A , aik P)
37
-
ni=1 pii =
ni=1
nk=1 akicki
tr((AT +BT )C) =ni=1
nj=1
((aji + bji)cji) =ni=1
nj=1
(ajicji + bjicji)
=ni=1
nj=1
ajicji +ni=1
nj=1
bjicji = tr(ATC) + tr(BTC) =
= A,C+ B,C
A,B = tr(ATB) =ni=1
nj=1
ajibji = ni=1
nj=1
ajibji =
= tr(ATB) = A,B
2 ():
A,B = tr(ATB) =ni=1
nj=1
ajibji =ni=1
nj=1
bjiaji =
= tr(BTA) = B,A
3:
A,A = tr(ATA) =ni=1
nj=1
ajiaji =ni=1
nj=1
a2ji 0
- aji = 0 i, j, A = O. Cauchy-Schwartz
| A,B | AB |tr(ATB)| tr(A)
tr(B)
|ni=1
nj=1
ajibji| n
i=1
nj=1
a2ji
ni=1
nj=1
b2ji
2.2.3 Rn[x] -
p(x), q(x) = 10
p(x)q(x)dx p(x), q(x) Rn[x]
.
Cauchy-Schwartz. p(x) = x2,q(x) = x 34 .
38
-
p(x), q(x), r(x) Rn[x] R. ,A1 ( ):
p(x) + q(x), r(x) = 10
[p(x) + q(x)]r(x)dx = 10
p(x)r(x)dx+
+ 10
q(x)r(x)dx = p(x), r(x)+ q(x), r(x)
p(x), r(x) = 10
p(x)q(x)dx = 10
p(x)q(x)dx = p(x), q(x)
2 ():
p(x), q(x) = 10
p(x)q(x)dx = 10
q(x)p(x)dx = q(x), p(x)
3
p(x), p(x) = 10
p2(x)dx 0
- p(x) = 0 [0, 1] x R ( n n ). Cauchy-Schwartz 1
0
p(x)q(x)dx
10
p2(x)dx
10
q2(x)dx
,
x2 = 1
0
x4dx =55,
x 34 =
10
(x 3
4
)2dx =
2112
p(x), q(x) = 10
x2(x 3
4
)dx = 0
( = pi2 ).
2.2.4 F2pi(R) - 2pi
f(x), g(x) = 12pi
2pi0
f(x)g(x)dx
.
Cauchy-Schwartz. {cosx, . . . , cosnx} {sinx, . . . , sinnx}
39
-
.
A1 A3 (2.2.3), Cauchy-Schwartz
(
[0, 2pi]) ,
cosmx, cosnx = 12pi
2pi0
cosmx cosnxdx ={
0, m 6= n1, m = n
cosnx =
12pi
2pi0
cos2 nxdx = 1
, {cosx, . . . , cosnx} . {sinx, . . . , sinnx} ( ).
2.3
2.3.1 x y 6=0,
Pry(x) =x,yy2 y
2.3.1 x,y 6= 0 E,
x (Pry(x))y
x (Pry(x)) ,y =x
(x,yy2 y
),y= x,y x,yy2 y,y =
= x,y x,yy2 y2 = 0
Gram-Schmidt
n E n E.
2.3.1 Gram-Schmidt
x1, . . . ,xn E .
40
-
1.
y1 = x1
2.
y2 = x2 (Pry1(x2)).
.
.
n.
yn = xn (Pry1(xn)) . . .(Pryn1(xn)) y1, . . . ,yn .
, o y2y1 (2.3.1). y3 = x3 (Pry1(x3)) (Pry2(x3))
y3y1 (2.3.1) y2y1. y3y2 (2.3.1) y3y1. yky1, . . . ,yk1,
yk+1 = xk+1 (Pry1(xk+1)) . . . (Pryk(xk+1)) yj 1 j k (2.3.1) yk. yi 6= 0: y1 = x1 6= 0, x1, . . . ,xn . y2 6= 0 y2 = x2 x1 = 0 x1,x2
y1, . . . ,yn , n .
,
2.3.2 -
.
2.3.1 (Parseval) x,y n- - E (x1, . . . , xn), (y1, . . . , yn) {e1 . . . , en},
xk = x, ek k = 1, 2, . . . , n
x,y =n
k=1
xkyk
x = n
k=1
x2i
41
-
.
2.3.1 R3 x1 = (1, 2, 3), x2 = (1, 0,1), x3 = (0, 1, 3).
y1 = x1 = (1, 2, 3).
y2 = x2 Pry1(x2) = (1, 0,1)(1, 0,1), (1, 2, 3)
(1, 2, 3)2 (1, 2, 3) =
= (1, 0,1) 214
(1, 2, 3) =(87,27,4
7
)
y3 = x3 Pry1(x3) Pry2(x3) =
= (0, 1, 3) (0, 1, 3), (1, 2, 3)(1, 2, 3)2 (1, 2, 3)(0, 1, 3),
(17 ,
27 , 47
)( 17 ,
27 , 47
)2(17,27,4
7
)=
= (0, 1, 3) 1114(1, 2, 3)
107
8449
(87,27,4
7
)=(16,1
3,16
) R3
e1 =y1y1 =
(114,
214,
314
), e2 =
y2y2 =
(421,
121, 2
21
)
e3 =y3y3 =
(16, 2
6,16
)
2.3.1
{e1, e2, . . . , en} = {(1, 0, . . . , 0), (0, 1, . . . , 0), . . . , (0, 0, . . . , 1)}
Rn Rn.
2.3.3 E n- V ,
E = V V
{e1, . . . , em} V , x E. , v = x, e1 e1 . . . x, em em
42
-
V ( ) z = x (v) V , u = u1e1 . . . umem V , x (v),u = x,u v,u =
= x, u1e1 . . . umem x, e1 e1 . . . x, em em,u =
=mk=1
uk x, ek mk=1
x, ek ek,u (2.6)
ek,u = uk (?) (2.6)
x (v),u =mk=1
uk x, ek mk=1
uk x, ek = 0
, x E v V w = x (v) V , x = v w, V V . (1.5.3)o V V E, , V V E, .
2.3.2
( )
. V E ( , V ).
2.3.4
E.
V , V - .
.
2.4
-
.
1.
x,y = x1y1 + 2x1y2 + 2x2y1 + 5x2y2 R2.
43
-
2. -
(2.1.3).
3. E
. 4 x,y = x y2 x (y)2. 2(x2 + y2) = x y2 + x (y)2. x y x y. x y x (y). x = y, (x (y))(x y)4. v,w 6= 0 E, to vw, .
w. [ = v2v,w
]5. (2.1.4).
6. Rn - x,y = nk=1 xiyi . Cauchy-Schwartz
Minkowski;
7. a : [
a 00 1
] [a 12 12 a
]8. (2.2.4).
9. (2.3.1)
10. R3
x,y = (x1 2x2)(y1 2y2) + x2y2 + (x2 + x3)(y2 + y3)
R3 Gram-Schmidt. [{
(1, 0, 0),(
218, 1
18, 0), (0, 0, 1)
}]
44
-
11. R2[x] {1, x, x2} (- , 2.2.3).[{
1,12(x 12
),5(6x2 6x+ 1)}]12. R4 , - x = (1, 1, 1, 1) .
x y = (1,1,1,1). x y? [
i = pi6 , P ry(x) =( 12 , 12 , 12 , 12) , x,y = arccos 14]13.
. x 5y 2z = 0. 2x y z = 0 [() ..
{(16, 1
6, 2
6
),(
25, 0, 1
5
)}]14. R4 ,
(1, 1, 0, 0), (0,1, 0, 2), (0, 0,2, 1)[..
{(12, 1
2, 0, 0
),(0, 0, 2
5, 1
5
),(
5370
, 5370
, 8370
, 16370
)}]15. R3 x 2z = 0 ( ) [..
{(25, 0, 1
5
), (0, 1, 0) ,
(15, 0, 2
5
)}]16. v = (3, 1, 2,4) R4 , x1 =(1, 2,1, 1), x2 = (0, 3, 1,1) ( ).17. V V ( V - E),
(V ) = V
45
-
Fourier
, 23
Minkowski, . , Minkowski
, 4
Minkowski, 36
, 36
, . ,
, 42
, 17
, 15
, 4
, 5
, . -
,
, 10
, 35
, 19
, 20
, 22
, 16
, 16
, 35
, 33
, 33
, 40
, 32
, 15
, 32
, . , -
, 13, 14
, 35
, 40
, 35
, 35
, . ,
, 25
, 3
, . -
,
, 22
, 35
, 35
, . -
,
, 43
, 10
-, 11
,
24
, 11
46
-
, 43
, 43
47