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5/21/2018 . -
1/94
,
, 2006
5/21/2018 . -
2/94
Copyright by E Kappos, 2006
.
LaTEX
kerkis.
, , 10/2006.
5/21/2018 . -
3/94
.
.
.
( .)
, -
,
, . ,
, , -
: -
. ,
: ()
(
) () ,
, -
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. 1
1,
.. ..!
5/21/2018 . -
4/94
iv
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( ) .
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-
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.)
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2. -
( Gauss),
, LU (
.).
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.
-
2 .
5/21/2018 . -
5/94
v
.
-
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, .
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, ,
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, . , -
,
( , ; .)
(
.)
,
.
, -
. ,
v, u, w, . . . , ai, bi, . . .
, , . . . , , c, . . . .
(+ )v, ca + b, (u + v)
.
.
, , .
v , v, v
v.
.
,
. , v .
5/21/2018 . -
6/94
vi
5/21/2018 . -
7/94
vii
1 1
2 3
2.1 . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 . . . . . . . . . . . . . . 7
2.3 . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 . . . . . . . . . . . . . . . . 16
2.4.1 . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.2 . . . . . . . . . . . . . . . . . . 20
2.4.3 R3 . . . . . . . . . . . . . . 25
2.5 . . . . . . . . . . 30
2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 33
3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 R2: . . . . . . . . . . . . . . . . . . 33
3.3 R3: . . . . . . . . . . . . 35
4 37
4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 . . . . . . . . . . . . . . . . . . . . . 40
4.2.1 . . . . . . . . . . . . . . . . . . . 43
4.3 ;
. . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4 . . . . . . . . . . . . . . . . . . . 49
4.5 , 52
4.6 : . . . 54
4.6.1 . 56
4.6.2 . . . . . . . . . . . . . . . . . . . . 56
4.6.3 Gauss . . . . . . . . . . . . . . . . . . 59
5/21/2018 . -
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viii
4.7 . . . . . . . . . . 62
4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5 65
5.1 . . . . . . . 65
5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3 . . . . . . . . . . . . . . . . . . 67
5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.5 . . . . . . . . . . 74
5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
I 77
: 79
.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
85
5/21/2018 . -
9/94
1
, ,
.
(-
) , -
.
(.. -
)
.
-
:
:
-
.
:
s1, s2, . . . , s30 m1, m2, . . . , m45. sicij
mj . Si , ;
Xj, j = 1, . . . , 45 , :
c11X1+ c12X2+ + c1,45X45 = S1c21X1+ c22X2+ + c2,45X45 = S2
c30,1X1+ c30,2X2+ + c30,45X45 = S30
5/21/2018 . -
10/94
2 1.
,
.
: , -
,
(inventory allocation and control.) -
,
.
(Linear Programming.)
:
(
Descartes), -
.
,
Rn.
:
.
,
, .
: -
( -
)
. -
:
, Googlec,
-
!
5/21/2018 . -
11/94
2
2.1
v , . -
( ,
.)
,
. -
0. , , . -
, . -
, . ,
v1 + (v2 +v3) = (v1 +v2) +v3. , :
v+ w= (x1, y1, z1) + (x2, y2, z2) = (x1+ x2, y1+ y2, z1+ z2).
v -
v( .) ,
v= (x,y,z) = (x,y,z).
: -
n- (n = 2 , n = 4...),
a= (a1, a2, . . . , an),
5/21/2018 . -
12/94
4 2.
a b n-, , ,
a + b= (a1+ b1, a2+ b2, . . . , an+ bn).
a= (a1, a2, . . . , an).
2.1. -
: R3 (
.) , -
( .) , ,
, . -
,
.
,
. 2.3.
2.1. F= R C. V F-- (.. F = R .. F = C)
v, w V v + wV v V, F, v V :
1. v, w V,v + w= w + v,2. v, w, u V,(v+ w) + u= v+ (w+ u),3. V 0, 0 + v= v
v V,
4. , F,v, w V,(v+ w) =v + w, ( + )v= v+ v, (v) = ()v,
5. 1 F, v V: 1v= v,6. 0 F, v V: 0v= 0.
2.2. 0 0.
5/21/2018 . -
13/94
2.1 5
. , :
0= 0v= (1 + (1))v= 1v+ (1)v= v + (1)v,
(1)v, v, v v V.
1
Rn: , n- n- . - 0 = (0, 0, . . . , 0)(n.) n. - R2 () R3 (
), n 3 .
Cn: n- n- v = (z1, z2, . . . , z n)(zi C i), , C
v= (z1, z2, . . . , z n).
n = 1 C. C ,
!
z= 0 w
w= z = w/z.
, ( )
( ): -
z C z:
C= {z, C}.
2.1. C ,
R C, .
1 - , .
5/21/2018 . -
14/94
6 2.
n = 2 , Cn , , -
2n(.. n = 2, C2 4.) -
,
Fourier
.
2.3.
-
!
: -
n(n >0).
p(x) =a0+ a1x+ a2x2 + . . .+ anx
n
(n+ 1)-v = (a0, a1, a2, . . . , an) Rn+1. , , p(x) + q(x) p(x) v.
Rn+1.
-
:
: -
,
: f, g , x f(x)+g(x) f(x). . : f x,f(x), .
, .
: ( )
( v = {a0, a1, . . . , an, . . .}) , - ( .)
5/21/2018 . -
15/94
2.2 7
2.2
0 -
R3: v - , v
L(v) ={v; R}. - R3,v1, v2. :
:= {1v1+ 2v2; 1, 2 R}.
.
(), v2=v1 R, = 0. .
R3 :
1. 0 (1, 2 = 0.)2. w < v1, v2 >, w < v1, v2 >(
1, 2.)3. w1, w2 , w1+w2 ( v= 1v1+2v2,
w= 1v1+ 2v2 1+ 1, 2+ 2.)
4. v < v1, v2 >, R, v < v1, v2 >(:1, 2.)
< v1, v2> .
2.2. W V V V V.
, v, w W, R, v + wv W.
. R3
R3,
.
2.1. v1, v2, . . . , vk - V. ,< v1, v2, . . . , vk >, V.
5/21/2018 . -
16/94
8 2.
.
v1
-. v2,
v1, v2
v1. : v1, v2 , v3, v1, v2, ( ) v3 < v1, v2>. :
2.3.
. v1, v2, . . . , vk , -
vk+1, v1, v2, . . . , vk, vk+1 vk+1 / . v1, v2, . . . , vk , k
< v1, v2, . . . , vk>.
-
, -.
:
2.4. v1, v2, . . . , vk 1, 2, . . . , k
F, , -
1v1+ 2v2+ . . .+ kvk =0.
v1, v2, . . . , vk . ,
v1, v2, . . . , vk , i.
! -
, -,
. , -
-, : (1, 2) = (0, 0)
1v1+ 2v2 = 0.
( , 2. v2=v1, = 1/2.)
,
,
:
5/21/2018 . -
17/94
2.2 9
2.1.
(0, 1, 0), (0, 0, 1)(0, 1, 1) R3;,
(1) (0, 1, 0) + (1) (0, 0, 1) + (1) (0, 1, 1) = (0, 0, 0), .
2.2. -
(1, 1, 0), (0, 1, 1) (2, 1, 1)? x, y,z i :
x(1, 1, 0) +y(0, 1, 1) +z(2, 1, 1) = (0, 0, 0) - -
:x + 0 + 2z = 0x y z = 00 + y+ z = 0
2.2. .
; ;
2.5. V (..). V,dim V, .
V.
0 - (.. .) ..
..
,
, -
.
:
2.2. -
. ,
..V .{b1, b2, . . . , bn} ..V n.
v V x1, x2, . . . , xn v= x1b1+ x2b2+ . . .+ xnbn.
5/21/2018 . -
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10 2.
x1, . . . , xn v {b1, . . . , bn}.
:
! .
2.6. Rn, n
e1 = (1, 0, . . . , 0), e2= (0, 1, 0, . . . , 0), , en= (0, . . . , 0, 1).
(
!) ( . -
Rn) , .
2.3. R3
b1 = (0, 1, 1), b2= (1, 0, 1), b3= (0, 0, 2).
. R3
v = (3, 1, 1) . , ,
b1, b2, b3, (1, 2, 3)
=
(0, 0, 0) 1b1+ 2b2+ 3b3=0.
,
(2, 1, 1 2+ 23) = (0, 0, 0), 2 = 0, 1 = 0 3 = 0, -. b1, b2, b3 .
x1, x2, x3
v= x1b1+ x2b2+ x3b3.
(3, 1, 1) = (x2, x1, x1 x2+ 2x3),
x2= 3, x1= 1 x3 =11+3
2 = 0, 5. -
v {b1, b2, b3} (1, 3, 0, 5) (3, 1, 1). .
5/21/2018 . -
19/94
2.3 11
2.1. v = (c1, c2, . . . , cn)
Rn
{e1, e2, . . . , en) c1, c2, . . . , cn.
2.4. -
. , -
( 2.2)
(;)
2.3
,
,
.
: ,
. ;
, S,
(s1, s2), , s1 s2 ,
,
, (s1 s2)s3, , s1 s2 S; ,
. , ,
V.2 , , , R2 R3. R
. R3
, v1 v2( .)
R2 -
! v1 = (c1, c2)
2 , ,
, :
(x1, x2, . . . , xn) (y1, y2, . . . , yn) = (x1y1, x2y2, . . . , xnyn).
5/21/2018 . -
20/94
12 2.
z1 = c1+ ic2, . , v2= (d1, d2), :
v1 v2= (c1c2 d1d2, c1d2+ c2d1),
z1z2= (c1+ ic2)(d1+ id2) = (c1c2 d1d2) +i(c1d2+ c2d1). .
, , , (z1z2)z3=z1(z2z3) , ..(z1+ z2)z3=z1z3+ z2z3.
4, ( - R4!)
, ,
,
.
, (v1, v2)v1 v2. , v1 v2 v3 (;)
,
. :
2.7. V, v1, v2 ,v1 v2 R V :
1. v1 v2=v2 v1( ),2. v v 0 v v= 0 v= 0( ),3. (v1) v2 = (v1 v2),4. (v
1+ v
2)
w= v1
w + v2
w().
-
.
Rn :
v1, v2 Rn, v1 = (x1, x2, . . . , xn),v2= (y1, y2, . . . , yn)v1 v2=x1y1+ x2y2+ . . .+ xnyn.
. Rn -
, n.
5/21/2018 . -
21/94
2.3 13
,
v v= x21+ x22+ . . .+ x2n = ||v||2.( -
n: , v =(x1, x2) -,||v|| =
x21+ x22, ,
.)
Rn. , R3
,
v1 v2 = 2x1y1+ 3x2y2+ x3y3, . -
( -
!) ,
v1 v2= 2x1y1+ 3x2y2+ x3y3 x1y2 x2y1. . -
: ,
, - ,
! , , -
,
. , , . 2.8. v1, v2 V ( ) v1 v2 = 0.
Rn,e1 = (1, 0, . . . , 0),e2= (0, 1, 0, . . . , 0), ... en= (0, . . . , 0, 1) : ei
ej = 0i
=j .
2.4. (1, 1, 3), (3, 0, 1) R3 :(1, 1, 3) (3, 0, 1) = (1)(3) + 0 + (3)(1) = 0.
-
;
(x,y,z) :
(1, 1, 3) (x,y,z) =x y+ 3z= 0, (3, 0, 1) (x,y,z) = 3x + 0 z= 0.
5/21/2018 . -
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14 2.
, z = 3x, y = x+ 3z = 10x.
, , :
L = {(1, 10, 3); R}.
(
.)
:
. ,
( ) . , ,
{e1, e2, . . . , en} Rn: v= (x1x2, . . . , xn) x1, x2, . . . , xn! (2.1)
2.9. {b1, b2, . . . , bn} V - ,
i =j , b1 bj = 0, , . ||bi|| = 1 i(. ||v|| =
v v).
2.3.
{b1, b2, . . . , bn
} V.
v V n v :
x1 = v b1, x2=v b2, . . . , xn = v bn.. {bi}ni+1 , v - ,v = x1b1 + x2b2 + . . . + xnbn. bi:
bi
v= bi
(x1b1+ x2b2+ . . .+ xnbn) =
n
j=1
xjbi
bj =xi,
{bi}.
, -
. ,
,
. , -
Gram-Schmidt. 10.
5/21/2018 . -
23/94
2.3 15
: -
v1 v2 =x1y1+. . . xnyn . ,
v2=v1, ,
v1 v2=v1 v1=||v1||2. , -. , v1 v2 , .
: -
( -, .)
( ) v1 b1, ,
v2 < b1, b2 >( .)
2.3. .
v1 (x1, 0, . . . , 0), !
v2 = (y1, y2, . . . , yn)
.
, y1 v2 b1( .) ,
v1 v2= (x1, 0, . . . , 0) (y1, y2, . . . , yn) =x1y1, .
, ,
:
2.2. v1, v2 -
Rn
v1 v2 = ||v1| | | |v2|| cos , (0 180).2.5. (3, 1, 1)(1, 2, 0) .
:
||(3, 1, 1)|| =
32 + 12 + (1)2 =
11,
5/21/2018 . -
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16 2.
2.1: . () 0 , ()
()
||(1, 2, 0)|| =
(1)2 + (2)2 =
5.
,
cos =(3, 1, 1) (1, 2, 0)
11
5=
511
5
= cos1
511
= 132, 4
2.4
-
R2 () R3 (-
).
,
.
, ,
! (x, y x, y,z) : , :
(x,y,z), ; ,
,
R3, x, yz( , ;)
( R2
R3) ()
( 0). ,
.
. n-, Rn - e1 = (1, 0, . . . , 0), e2= (0, 1, 0, . . . , 0)...
5/21/2018 . -
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2.4 17
R3 ,
e1, e2, e3 i,j, k. v R3, v= xi + yj + zk,
x, y,z v. , -
, i,j, k . , (i,j, k). . (j, k, i)(k, i,j). , (j, i, k) -.
2.4.1
.
e1 = (1, 0), e2 = (0, 1). v = xe1+ye2 v = (x, y) R2 (. 2.1).3 .
v1= (x1, y1)v2 = (x2, y2)
v1
v2 = x1x2+ y1y2
||v|| =
x2 + y2.
e1 e2
( e1
e2.) (e1, e2) . - (b1, b2) b2 b190 ( .)
2.3. v = (x, y) - , v = (y, x) v (v, v) R2.
.
||v|| = 1( , ||v||.).
3
(x1, x2, . . . , xn) Rn, (x, y) .
5/21/2018 . -
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18 2.
.
(ABC) A = v1, B = v2 C = v3. AB , ,
|AB| = ||v2 v1|| =
(x2 x1)2 + (y2 y1)2.
, -
:
2.4. v1, v2 - , v1 = (x1, y1), v2 = (x2, y2), ,
x1y2 x2y1.
a b: E=ab sin , ( 90) .. -
, : v1, ,
(x1, 0) , v2 = (x2, y2), y2 x1y2( x2y1 = 0.)
A
B
C
x1
x2
y2
y1
x1+ x
2
C/
B/
0
2.2:
, .
v1, v2 , , A, B,C v1, v1 + v2, v2 , 0ABC.
5/21/2018 . -
27/94
2.4 19
x1CB(x1 + x2) x1CB(x1 + x2) ( 0CC ABB .) :
E(x1CB(x1+ x2)) =1
2(y1+ (y1+ y2)) x2 = y1x2+
1
2x2y2
E(x1CB(x1+ x2) =
1
2
x1x2
y2+ y2+x1x2
y2
x2=x1y2+
1
2x2y2.
( Cx1 .) . , , .
:
, -
e1
.
.
v = (x, y) = xe1+ ye2 .
,
x= v e1= ||v||||e1|| cos = ||v|| cos , e1 . r= ||v||, x= r cos .r2 = x2 +y2 y2 = r2 x2 = r2(1 cos2 ) = r2 sin2 . -,y = r sin . v -.
0 < 2( 0 .)
x= r cos , y = r sin (0 0, - .
2.4. v =0 (r, ), - - (r, ), (0, )[0, 2).
5/21/2018 . -
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20 2.
2.5. ,
: (r, ) .
.
, , 0
r.
2.4.2
R3. e1 =(1, 0, 0), e2 = (0, 1, 0) e3 = (0, 0, 1)
i,j k e1 = i, e2 = j, e3 = k. v =(x,y,z)( ) v= xi + yj + zk. x, y,z v . {i,j, k} .
(
: ,
.)
xi, yjzk v -
/ ( .) -
v
||v|| =
x2 + y2 + z2,
v1 v2= (x1, y1, z1) (x2, y2, z2) =x1x2+ y1y2+ z1z2 R:||v|| = v v.
:
v1 v2= ||v1||||v2|| cos , ( [0, ]) . v :
v i= xi i + yj i + zk i= xi i= x,i j= 0, i k= 0. , ,
x= v i= ||v||||i|| cos 1 = ||v|| cos 1,
5/21/2018 . -
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2.4 21
1 vi. ,
y=v j= ||v|| cos 2 z=v k= ||v|| cos 3,2, 3 v j, k. - v
, :
||v||2 =x2 + y2 + z2 = ||v||2(cos2 1+ cos2 2+ cos2 3), :
cos2 1+ cos2 2+ cos
2 3 = 1
2.6.
: (1, 2, 3, r = ||v||). ,
. ,
(1, 2, r) (-) R3. : 2 1!
2.4. |2| cos1
1 cos2 1. 1 =.
2= ; .
, -
.
R3: -
,
: ,
, -
. ,
, -
. ,
i
Greenwich .
r= ||v|| . . :
2.10. R3 {i,j, k}, -v, :
5/21/2018 . -
30/94
22 2.
1. kv(v
k=
||v
||cos ),
2. i v < i,j >, vzk, [0, 2).
3. r= ||v||, v. (,,r) - v
(x,y,z)
v= xi + yj + zk :
x = r sin cos y = r sin sin z = r cos
-
.
2.5. R3
( < k > k, ) (0, ) [0, 2) (0, ), (,,r).
k(-
0), -
. (x,y,z)
= cos1(v
k
||v||), = tan1
y
x, r= x2 + y2 + z2.
. , x2 +y2 = 0(- < i,j>.)
=
tan1 yx
x 0, y 0+ tan1 y
x x 0
tan1 yx
+ x 0, y 02+ tan1 y
x x >0, y
5/21/2018 . -
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2.4 23
,
, . ,
.
R3
r( ), ,
.
, , r ,
. , ,
, r = 0 x = y = 0 .
R3: -
< i,j >, z,
= x2 + y2, = tan1 yx
, z.
, -
(,,z)
x = cos y = sin z = z
= 0, , , - .
2.6. R3 < k > (0, )[0, 2)R, (,,z).
= z = =.
5/21/2018 . -
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24 2.
R3: .
, -
: (, ) (, +0). -
.
,
.
, -
. , .
,
.
2.11. T V W , v1, v2V 1, 2 R,
T(1v1+ 2v2) =1T(v1) +2T(v2).
R3
R3.
.
2.12. R3, -
(rotation) , -
( , ) , -
.
Rot : R3 R3. R3 - SO(3).
-
. ,
:
2.13. T :R3
R3
,
T(v1) T(v2) =v1 v2. : -
, -
, .
. O(R3). .
5/21/2018 . -
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2.4 25
:
, ,
(,,r) r, -(+0, +0, r), 0, 0, , r. , , 00 - , 0, . -
, . ,
(,,r) (, +0, r) (,,r) (+0, , r) . -, ,
r 0, - ,
,
! , -
, Roti, Rotj, Rotk. .
-
: i
( ). -
!
,
/ SO(3) .
2.4.3 R3
.
. -
.
,
.
-
.
2.14.v1 = (x1, y1, z1), v2 = (x2, y2, z2)
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26 2.
, v1
v2
v1 v2 = (y1z2 z1y2)i + (z1x2 x1z2)j + (x1y2 y1x2)k (2.1)
2.6. (2, 1, 4) (1, 1, 0)
(2, 1, 4) (1, 1, 0) = (4)i + (4)j + (2 1)k
:
1. :
v (v1+ v2) =v v1+ v v2
v1 (v2) = (v1) v2=(v1 v2),0 v= 0.
2.
v
v= 0
3. ,
v1 v2= v2 v1( (v1+ v2) (v1+ v2) = 0, .)
(1) (2)
(;)
.
2.5. v1 = (x1, y1, z1), v2 = (x2, y2, z2) , n
v1, v2 (v1, v2, n) R3. ,
v1 v2= (||v1||||v2|| sin ) n,
(0 ).
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2.4 27
||v1
||||v2
|| |sin
|
v1 v2.
, , v1 v2, ,
v1, v2 - .
, ,
( .)
, -
F r,F r.
. (u,v,w) v1= (x1, y1, z1),v2= (x2, y2, z2)
x1u + y1v+ z1w = 0
x2u + y2v+ z2w = 0
w ,4
u = wy1z2 z1y2x1y2 y1x2
v = w
z1x2
x1z2x1y2 y1x2
, w = x1y2 y1x2, u = y1z2 z1y2v = z1x2 x1z2.
(y1z2 z1y2)i + (z1x2 x1z2)j + (x1y2 y1x2)k
v1 v2.
v1 v2. ||v1||||v2|| | sin | . ,
, -
( ,
.)
-
:
4 x1y2y1x2 -. , ,
.
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28 2.
v1 v2 =
i j kx1 y1 z1x2 y2 z2
( .)
2.7.
(3, 1, 0) (2, 1, 1) =
i j k
3 1 02 1 1
= (1)i + (3)j + (3 + 2)k
:,
(v1 v2) v3=v1 (v2 v3)
.
2.5.
(v1 v2) v3=0, v1 (v2 v3) =0; 2.6.
1.(v1 v2) v3+ (v2 v3) v1+ (v3 v1) v2=0
2.
(v1 v2) v3= (v1 v3)v2 (v2 v3)v1,
:
v1 v2 v3 = v1 (v2 v3)
: (v1 v2) v3 ! ;
:
2.7. v1, v2, v3 ,
.
2.6. -
.
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2.4 29
:
:
v1 v2 v3=
x1 y1 z1x2 y2 z2x3 y3 z3
. ,
,
F Stokes.
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30 2.
2.5 -
: -
( R C.)
.
.
: n - n. -
. . ,
.
:
.
, .
R2 R3
((i,j)(i,j, k))
R3 :
(x1, x2, x3) (y1, y2, y3) =x1y1+ x2y2+ x3y3 R.
: -
,
. :
R3 -
.
: ,
, -
.
:
v1 v2 =
i j k
x1 y1 z1x2 y2 z2
.
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2.6 31
2.6
1. -
.
2. R4
v= (1, 3, 0, 4), w= (2, 0, 3, 1). 2v w v 3w.
3. C2
v= (1 +i, 1 + 2i), w= (2i, 1 i) v+ (2i)w(1 + 2i)v.
4.
{(1, 0), (0, 1)}, {(i, 0), (0, i)}, {(0, 1), (i, 0)} C2.
5.
(1, 2, 1), (2, 1, 0) (1, 0, 0) R3.
6. v V {b1, b2, . . . , bn} .
7. (2, 2, 4) b1 = (1, 2, 0),b2= (1, 1, 1)b3 = (0, 2, 3).
8. ;
() v1, v2, . . . , vk , (.. v1, v2, . . . , vk1) -.
() v1, v2, . . . , vk , (.. v1, v2, . . . , vk1) -.
() {b1, b2, . . . , bn}{c1, c2, . . . , cn} -V. bi - {c1, c2, . . . , cn}.
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32 2.
9. n Rn+1. -
p(x), q(x),p(x) q(x),
.
Rn+1 ( n) ; n = 1 ;
10. Gram-Schmidt: {b1, b2, . . . , bn} V. e1 = b1/
||b1
||.
e2, e3, . . . en , e1, V.
E2=b2(b2 e1)e1 e1. - e2=E2/||E2||. - : E3=b3 (b3 e1)e1 (b3 e2)e2. E2 e1 e2. e3=E3/||E3|| , e4, . . . , en V. - Ei
.
11. v (1, 2, 1) (0, 1, 1). - ,
R3.
12. e1= (1, 0, 0), e2= (0, 1, 0), e3 = (0, 0, 1)R3.
b1= (cos , sin , 0) b2 = ( sin , cos , 0)
.
b1 b2 b1, b2.
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3
3.1
( 16 ) -
,
.
(Rene Descartes, 15961650) -
.
,
-, .
,
,
.
.
3.2 R2:
:
L = {(x, y) R2; ax+ by= c}, a,b,c R.
b = 0,
y=x+ , (= a/b, =c/a.)
b = 0 . , (
5/21/2018 . -
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34 3.
)
n0. n0 = (a, b) ax+ by (x, y) n0:
n0 (x, y) =ax+ by= c. , : (x, y) (x1, y1) , n0:
(x x1, y y1) (a, b) = 0.c = 0 R2. c
= 0,
(affine).
(x, y) !,
! -
, ,
. , (x, y) :
ax+ by = cx+ dy =
3.1. ax +by = cx +dy =( ), ,
(c,d,) =(a,b,). , (c, d) =(a, b).
, ,
.
.
:
v0= (x0, y0) R >0 () R v0 :
C(v0, R) = {v R2;|v v0| =R}. , :
C(v0, R) = {(x, y) R2; (x x0)2 + (y y0)2 =R2}., :
x2 + y2 + dx+ ey = f, d= 2x0, e= 2y0, f=R2 x20 y20.
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3.3 R3: 35
3.1.
, x0, y0, R;
: (x, y) :
ax2 + by2 + cxy+ dx + ey = f .
,
. ( -
), , ,
...
3.3 R3:
: n0 = (a,b,c)= 0, (x,y,z)- n0 .
ax+ by+ cz= 0,
.
, :
ax+ by+ cz=.
R2 ,
, n0.
:
3.1. (1, 1, 2) (0, 3, 5).
(x,y,z)
(x 0, y (3), z 5) (1, 1, 2) = 0.
x + y+ 2z= 3 + 10 = 7.
.
.
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36 3.
, , .
. -
R3
. , -
: , ,
,
.
.
(
4. , , -
!
4.
:
R3 -
f(x,y,z) =ax2 + by2 + cz2 + 2dxy+ 2eyz+ 2f xz+ gx+ hy+ iz
. ,
. ,
Q b. :
Q=
a d fd b e
f e c
, b= gh
i
.
, x = xi + yj + zk :
f(x) =xTQx + bTx= x
Qx + b
x,
,
-
. Q ,QT =Q. (level sets)
, R, f1() = {x R3; f(x) =}.
[ ]
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4
4.1
.
:
, !
: , , Heisenberg,
,
Lie .
4.1. m n m n m n .
A=
a11 a12 a1na21 a22 a2n
am1,1 am1,2 am1,nam1 am2 amn
(4.1)
M(m, n).
4.1. ,
, -
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38 4.
.1 , -
aij R C. M(m, n; R)M(m, n; C).
, -
. ,
-
, :
n, , n, n ! , , - ,
n n
n
, A M(n, n; R)! A = (aij)
m,n
i=1,j=1 = (aij), . -
:
am1,n2 am1n2.
(column vector) -
a, A, A M(m, 1). ,
. , -
, (rowvector). ,
. ,
A M(m, n) n , m. a1, a2, . . . , an A( , m), :
A= (a1 a2 . . . an) .
4.1.
A= 2 3 18 1 112 3 3
:
a1 =
28
2
, a2 = 31
3
, a3 = 111
3
.
1 : ,
, , ,
.
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4.1 39
4.2. AT A
M(m, n) -
M(n, m) A A . ,
A=
a11 a12 a1na21 a22 a2n am1 am2 amn
AT
=
a11 a21 am1a12 a22
am2
a1n a2n anm
, -
:
(c1, c2, . . . , cn) =cT, c=
c1c2
cn
, A M(m, n) - m n:
A=
aT1aT2. . .aTm
,
aT1 = (a11, a12, . . . a1n), aT2 = (a21, a22, . . . a2n) ...
4.2. ,
A=
2 3 18 1 11
2 3 3
, :
a1=
23
1
, a2 = 81
11
, a3 = 23
3
.
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40 4.
4.2
,
, .
.
4.3. A B M(m, n)( ), , A+ B :
A+ B=
a11 a12 a1na21 a22 a2n
am1,1 am1,2 am1,nam1 am2 amn
+
+
b11 b12 b1nb21 a22 b2n
bm1,1 bm1,2 bm1,nbm1 bm2 bmn
= (4.2)
=
a11+ b11 a12+ b12 a1n+ b1na21+ b21 a22+ b22 a2n+ b2n
am1,1+ bm1,1 am1,2+ bm1,2 am1,n+ bm1,n
am1+ bm1 am2+ bm2 amn+ bmn
, A+ B= (aij+ bij).
AB , .
4.4. A- A
A=
a11 a12 a1na21 a22 a2n
am1,1 am1,2 am1,nam1 am2 amn
(4.3)
A B M(m, n)A + (1)B 1A1+ 2A2+ + kAk, Ai M(m, n) i = 1, 2, . . . , k.
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4.2 41
: ,
:
, A B aij bij . , , ,
(. 4.3 .)
:
.
m nx1, x2, . . . , xn
a11x1+ a12x2+
+ xna1n = b1
a21x1+ a22x2+ + xna2n = b2
am1x1+ am2x2+ + xnamn = bn(4.4)
, -
(x1, x2, . . . , xn) , bi. , A= (aij)
x=
x1
x2 xn
, b= b1
b2 bn
Ax= b
Ax , : A -
x m b.
A B . , A B, A B.
4.5.AB, A M(m, n)B M(n, k). A m
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42 4.
( ):
A=
a11 a12 a1na21 a22 a2n
am1,1 am1,2 am1,nam1 am2 amn
=
aT1aT2 aTm
ai=
ai1 ai2 ain
, i= 1, . . . , m
B k :
B
b11 b12 b1kb21 a22 b2k
bn1,1 bn1,2 bn1,kbn1 bn2 bmk
=
b1 b2 bk
,
bj =
b1jb2j bnj
, j = 1, . . . , k .
, AB , (m, k),
AB=
a1 b1 a1 b2 a1 bka2 b1 a2 b2 a2 bk
am b1 am b2 am bk
(4.5)
, i j AB
(AB)ij =ai
bj =
n
l=1
ailblj .
4.3.
2 3 13 0 2
1 1 2 11 0 2 2
2 3 0 1
2 42 + 3 + 2 2 + 0 3 4 6 + 0 2 + 6 + 13 + 0 + 4 3 + 0 6 6 + 0 + 0 3 + 0 + 2
=
7 1 2 97 3 6 5
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4.2 43
: B A , B4, A2.
4.2.1
, , -
. , , -
12 -
,
A= (A1A2 A3),
, A1 , A2 A3 .
, -
(partition) A M(m, n):
A=
A11 A12 A1kA21 A22 A2k Al1,1 Al1,2 Al1,kAl1 Al2 Alk
,
Aij , (mi, nj). m:
m= m1+ m2+ . . .+ ml
n:
n= n1+ n2+ . . .+ nk.
A=
B CD E
,
B, D C, E B, C D, E .
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44 4.
: -
, . -
.
4.4.
2 21 3
1 45 3
0 00 0
2 21 1
1 11 2 2 12 00 00 0
3 12 1
4 22 7
11 329 9
0 00 0
10 41 0
-
. .
4.1. -
.
4.3 ;
: ,
() ,
a11, . . . , a1n, a21, . . . a2n, . . . , am1, . . . , amn.
;
.
; ,
.
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4.3 ;
45
2. V n -, n b1, . . . , bn , < b1, . . . , bn > V. - : B ={b1, . . . , bn}. v V
v= x1b1+ . . .+ xnbn (4.6)
x1, . . . , xn -v B.
v :
x1x2...
xn
. (4.7)
-
! , .
:
x1x2...
xn
B
. (4.8)
4.6 4.8 .
, ,
,
( ,
2.3.) V. E ={e1, . . . , en}
, :
1. , bi,i = 1, . . . , n, E= {e1, e2, . . . , en},
bi=a1ie1+ a2ie2+ . . .+ anien
( e1, . . . , en .)
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46 4.
2. E, ei, i = 1, . . . , n, B,
ei = 1ib1+ 2ib2+ . . .+ nibn
( b1, . . . , bn .)
v E= {e1, . . . , en}:v= x1(a11e1+ a21e2+ . . .+ an1en) +. . .+ xn(a1ne1+ a2ne2+ . . .+ annen)
v= (a11x1+ a12x2+ . . .+ a1nxn)e1+ (a21x1+ . . .+ an2xn)e2+ + (an1x1+ . . . annxn)en = y1e1+ y2e2+ + ynen (4.9)
, : -
Aaij( , m = n.) v B -
4.8. v E
y1
y2...
yn
E
=A
x1
x2...
xn
B
, (4.10)
, ,
vE= AvB (4.11)
, -
E B,
A:
vB= A1vE (4.12)
.
4.2. A, B
E. , .
,
.
.
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4.3 ;
47
4.5. R2: 10
,
01
,
11
,
11
,
E= {e1, e2} B= {b1, b2} . :
E B
b1 = e1+ e2b2 = e1+ e2
A
A=
1 11 1
.
4.12,
B A1,
A1 =1
2
1 11 1
.
2e1+ 0e2, ,
e
b
b
e
1
2
2
1
4.1: .
A1
20
=
11
B
.
: , ,
.
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48 4.
: T V W- 1v1+ 2v2
1T(v1) +2T(v2).
, v1, . . . , vn V
v= x1v1+ x2v2+ + xnvn, , v T
T(v) =x1T(v1) +x2T(v2) +
+ xnT(vn).
, T(vi) vi V W, w1, . . . , wm:
T(vi) =a1iw1+ a2iw2+ + amiwm.,
T(v) =x1(a11w1+a21w2+ +am1wm)+x2(a12w1+a22w2+ +am2wm)+ + xn(a1nw1+ a2nw2+ + amnwm). (4.13)
W,
T(v) = (x1a11+x2a12+ +xna1n)w1+(x1a21+x2a22+ +xna2n)w2+ (x1am1+ x2am2+ + xnamn)wm. (4.14)
A
A=
a11 a12 a1na21 a22 a2n
am1,1 am1,2 am1,nam1 am2 amn
T v1, . . . , vn Vw1, . . . , wm W.
4.1. 1. x -
v v1, . . . , vn, T(v) w1, . . . , wm
y= Ax
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4.4 49
2. A V,
A=
a1 a2 an
,
ai=T(vi).
,
ai=a1iw1+ a2iw2+ + amiwm.
4.4
, ,
,
.
4.6. 1.
, m = n.
2. -
, aij = 0 i= j. A =(a1, a2, . . . , an), ai = aii, i =1, . . . , n .
3. n 1, , In, , , InA= A,BIn=B.
4.
, aij = 0 j > i. .
5. A i, j,aij =aji,
A=
a11 a12 a1na12 a22 a2n a1n a2n ann
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50 4.
6. A i, j,aij = aji,
A=
0 a12 a1na12 0 a2n
a1n a2n 0
7. A Toeplitz( ),
A=
a0 a1 a2 an1a1 a0 a1 an2a2 a1 a0 a1
an+2 a1 a0 a1an+1 an+2 a2 a1 a0
Toeplitz
, .
8. A Vandermonde,
A=
1 1 1 11 2 3 n21
22
23 2n
n11
n12
n13 n1n
: 1, 2, . . . , n, Vandermonde .
1i j, .
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4.4 51
4.4. , A = AT.
: Am n Bn k, AB. , n 1, M(n, n) M(n), .
4.1. M(n) - ,
.
(AB)C=A(BC)( ). M(n) In. ,
(n = 2, ): 1 10 0
0 10 1
=
0 20 0
0 10 1 1 1
0 0 = 0 0
0 0
4.7. A M(n) B M(n) AB=BA= In. B A B = A1. M(n, n) Gl(n).
4.2. Gl(n) .
: , -
AB , A, B Gl(n) B1A1. 4.8. -
,
AAT =ATA= I .
, ,
(
(4.5).
M(n) n 2, :
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52 4.
4.6.
A=
1 20 0
: B, AB .
M(n) , , : A(B+C) =AB+ AC (A+B)C= AC+BC. M(n)(-).
4.7.
A=
1 00 2
A1 =
1 00 1/2
4.2. A ,A = (a1, . . . , an)
ai= 0, A1
A1 =(1/a1, . . . , 1/an).
4.2.
: A - ,
, .
,
!
.
, .
4.5 ,
A VW, -. , A
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4.5 , 53
V:
A=
a1 a2 an
,
ai= T(vi) W.
4.9. < a1, a2, . . . , an > W A W (image) A Im(A). (rank) A rank(A).
A, .
4.10. (kernel) A,ker(A), V ,
ker(A) = {v V; Av= 0}.
ker(A)V: ker(A) V. A A,
A.
rank(A) max{m, n}.
4.3. A M(n) :
1. n: rank(A) =n
2. -: det A = 0.3. W: Im(A) =W
4. : ker(A) = {0}.
( .)
.
A .
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54 4.
4.11. A M(n). (adjoint matrix) A A:
AdjA=
11 12 1n21 22 2n n1 n2 nn
T
=
11 21 n112 22 n2 1n 2n nn
(4.15)
4.3. A M(n) , A,A1
A1 = 1
det AAdjA
.2.
4.8.
1 1 22 0 10 2 1
|A| = 8 + 2 + 2 = 4 = 0, A .
:
11= 2 12= 2 13= 421= 3 22= 1 23 = 231= 1 32= 3 33 = 2
( !) ,
A1 = 1
4 2 3 1
2 1 3
4 2 2 4.6 : -
. -
!
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4.6 : 55
m n
a11x1+ a12x2+ + xna1n = b1a21x1+ a22x2+ + xna2n = b2
am1x1+ am2x2+ + xnamn = bn
(4.16)
, , ,
Ax= b.
4.4 ( ). A
A=
a1 a2 an
.
(4.17) b
ai:
b=n
i=1
xiai
,
. ba1, . . . , an m- R
m.
-
n b.: Rm, Rn!
:
4.4. A Im(A) =< a1, . . . , an > = Rm,
Rm, .
m= n, Im(A) =Rn a1, . . . , a
n. , ,
.
:
Rm, W Rm; :
4.5. Im(A) ==W=Rm, Rm, (4.17) b W.
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56 4.
4.6.1 -
- m = n, :
4.6.
a11x1+ a12x2+ + xna1n = b1a21x1+ a22x2+ + xna2n = b2
an1x1+ an2x2+ + xnann = bn(4.17)
:
1. det A = 0.
x= A1b
2. det A= 0, b A, :
b / .
3. det A= 0 b A:
b .
.
4.6.2
a11x1+ a12x2+ + xna1n = 0a21x1+ a22x2+ + xna2n = 0
am1x1+ am2x2+ + xnamn = 0
Ax= 0 (4.18)
: -
, 0. :
; .
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4.6 : 57
4.9.
2x + y = 06x+ 3y = 0
(4.19)
0= x
26
+ y
13
.
:
26
= 2 1
3
.
, (x, y) = (0, 0), -(x, y) = (1, 2), R . :
y = 2x. !
( , ),
, :
4.5. -
Rn,
A. m = n, - A .
dim ker A= n rank(A) . .
.
,
, R3.
4.10.
2x y+ z = 00 + 2y z = 0
4x + 2y+ 0 = 0(4.20)
:
det
2 1 10 2 14 2 0
= 4 8 + 4 = 0
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58 4.
: - . .
( ) :
z= 2y, x= y/2.
y =, z= 2,x = /2. A, (1/2, 1, 2):
ker A=< 1/21
2
>= { 1/212
; R}. 4.7( ). A k (k min{m, n}), k xi1, xi2 , . . . , xik k , xi1, . . . , xik ( ),
A
xi1
xi2 xik
= b({xj}j=i1,i2,...,ik),
A b - .
k , .
(. 4.6.1)
xi1
xi2 xik
= A1b({xj}j=i1,i2,...,ik)
, n k , Rn, A.
-
, , ,
,n
k .
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4.6 : 59
4.11 (). ,
0 z = 2y4x+ 0 = 2y , A=
0 14 0
z = 2y, x =y/2, y, .
4.6.3 Gauss
Gauss . -
,
.
.
,
Gauss
.
Gauss(Gauss Tableau).
Ax= b.
:
1. .
2. ( -
.)
3. A.
4. b.
5. .
, , / Gaussm:
N A b 1 a11 a12 a1n b1
ni=1 a1i+ b1
2 a21 a22 a2n b2n
i=1 a2i+ b2...
... ...
...
m am1 am2
amn bm ni=1 ami+ bm
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m1() :
,
A. . .
N A b 1 a11 a12 a1n b1
ni=1 a1i+ b1
2 2 a21 a22 a2n b2n
i=1 a2i+ b2...
... ...
... ...
m m am1am2 amn bm ni=1 ami+ bmm+ 1 0 a(m+1)2 a(m+1)n bm+1 ni=1 a(m+1)i+ bm+1...
... ...
...
2m 1 0 a(2m1)2 a(2m1)n b2m1n
i=1 a(2m1)i+ b2m1
: -
...
.
, , -
, : -,
. -
(pivots).
. -
!
(
A, b) ( .
4.12. 3 3:N A b 1 2 2
1 1 4
2 1 3 2 1 13 1 1 2 2 4
:
N A b 1 2 2 1 1 42 1/2 1 3 2 1 13 1/2 1 1 2 2 44 0 4 2, 5 1, 5 35 0
2
1, 5
2, 5
6
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4.6 : 61
, :
N A b
1 2 2 1 1 42 1/2 1 3 2 1 13 1/2 1 1 2 2 44 0 4 2, 5 1, 5 35 1/2 0 2 1, 5 2, 5 66 0 0 2, 75 1, 75 4, 5
1,4 6:
(6) = z= 2, 25/ 2, 75 = 0, 637, (4) = y=0, 5 + 2, 5z4
= 0, 772,
(6) = x= 1 2y+ z2
= 0, 047.
LU A (LU-decomposition). , ,
-
.
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62 4.
4.7 -
: -
( : cA+ dB). AB A B. A AT .
.
:
. A
.
M(n, n; R): . , A: A= AT A= AT,AAT =I,Toeplitz Vendermonde
n 1 i. : A M(n, R) det A= 0,
A1 = 1
det AAdjT(A).
det A = 0, A : Av = 0. A ,
- -
.
: Ax = b A b,
b .
ker A A,
A.
Gauss , -
.
Gauss.
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4.8 63
4.8 1.
A=
3 0, 3 1, 40, 4 0 2, 7
1, 3 2, 5 4
, B= 0 1, 2 8, 30, 7 3, 3 5, 1
0 7 6, 2
.
A 2B,A2,3A+ B2,(A B)(B+ A).2.
A=
1 3 44 0 7
, B=
3 51 2
0 4
.
, -
, .
AB, BA, BT 2A, AAT 3BTB, BBT A, BBT + ATA.
3.
, ,
A1 = 1|A|AdjT(A).
A=
1 2 22 0 3
1 2 4
, B= 0 2 37 3 5
3 1 6
.
4. ,
Gauss.
5. .
()2x y+ 3z = 0
3x 2y+ z = 12x+ y 2z = 2
()
0 + 4y z = 103x + y+ 3z = 3
x + 0 4z = 0
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64 4.
()2x y+ 3z = 0
3x 2y+ z = 12x + y 2z = 2
6. .
()2x 3y+ z = 2
x 2y+ 3z = 1()
2x + y 2z = 1x + y z = 1
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5
5.1 -
.
A - Rn
Rn. A , , -
. v , x1, x2, . . . , xn , Av =
ni=1 xiai
v.
v
v,
Av= v (5.1)
; , , -
Rn!
, , (
) .
(
,
.) , , -
(5.1). n v ,(n+1) . n! , . .
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66 5.
- (5.1) :
, v (5.1).
5.2
(5.1) :
(A
In)v= 0 (5.2)
In Inv = v. , , ( 4.6.2).
- A I n.
det(A I) = 0 (5.3) .
5.1. (5.3) , n.
n , - C.
A n A.
5.1. 1 12 5
det(A I2) = 1 12 5 = ( + 1)( 5) + 2 = 2 4 3.
= 2 7.
: 1 2 7 12 5 2 7
=
3 7 12 3 7
(
72
32) + 2 = 0.
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5.3 67
5.2. 3 1 30 1 2
0 0 4
det(A I3) =
3 1 30 1 20 0 4
= ( + 3)( 1)( 4)
=
3, 1, 4, .
:
5.2.
: 1 =a11, 2 =a22, . . . , n=ann.
, -
n -n . n.
k , k.
5.3
, (AIn) n . - .
5.1. - v Rn (A
I)v= 0
. .
5.1. : v
, c R, cv - (
Rn). , ,
.
-
.
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68 5.
5.3. 3 0 20 3 0
0 0 2
= 3, 3, 2. = 3,
A I3=
0 0 20 0 00 0 1
, ker(A+3I), , !
ker(A+ 3I) =< e1, e2 >
(e1, e2 R3.)
= 3 . -
, .
5.1. A, 1, 2, . . . , n (i= j i= j), i Vi ()
. 1 Rn:
< V1, V2, . . . , V n > = Rn.
:
5.4.
10 10 2 1= 102 = 2( .)
1= 10, (A 1I)v= 0:
(A 1I)v=
0 10 8
uv
=
00
.
1 . :
.
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5.3 69
: v = 0 u! i, 1 = 10 . :
v1 = c
10
.
,2 = 2:
(A 2I)v= 8 1
0 0 uv
=
00
.
: 8u + v= 0 :
v2 = c
18
.
5.5. 3 12 2
:
det(A I) = ( 3)( 2) + 2 = 2 5 + 8 = 0
=5
21
2
25 32 =5
2 i
7
2 C.
, -
:
(A
1I)v=
12 i
72
1
2 1
27
2 u
v = 0
0 .
:
v=
1
2 i
7
2
u.
, :
v= c 112
i72 ,
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70 5.
c C! !
2u=
1
2+
7
2
v=
1
2+
7
2
1
2 i
7
2
u=
1
4+
7
4
u= 2u!
, A I -
( .)
-
M(n; R) M(n; C),
! -
( n n):5.3. C A M(n; R) () = 0, , , - .
Av= v:Av= v,
A . , :
5.1. v2 =52 i
72 .
-
. ,
-
. , ,
Jordan.
5.2. k > 1, k , -: v1, . . . , vm (m k) - w1, . . . , wkm. vi /Avi= vi.
w -
Aw= w + v (A I)w= v, (5.4)
v .
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5.4 71
:
1. .
k, (. .)
2. , vi -
: (5.4) v= vi .
v . ...
.
3. , , - .
-
k, .
5.4
: A M(n), - , , ,
, !
. ,
, 1, 2, . . . , n, i= j i= j, v1, . . . , vn R
n (
.)
, , -
A . A ( 4.3.)
U= {v1, . . . , vn}. ( 4.3
):
U= (v1, . . . , vn)
v, U:
v= y1v1+ y2v2+ . . .+ ynvn.
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72 5.
A :
Av= A(y1v1+ y2v2+ . . .+ ynvn) =
=y1Av1+ y2Av2+ . . .+ ynAvn=y11v1+ y22v2+ . . .+ ynnvn. (5.5)
, U, A ,
=
1 0 00 2 0 0 0 n
.
, A , !( .) -
, ,
A, M(n; C)!
A . - U :
vU=
y1y2...
yn
U
vE=
x1x2...
xn
E
,
, , , :
vE= UvU.
v= x1e1+ x2e2+ . . .+ xnen=y1v1+ y2v2+ . . .+ ynvn.
x = vEy = vU, x = Uy. :
Ax= A(Uy),
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5.4 73
5.5. Ax v
. , U,
y. , :
Ax= U(y)
( y .) x =Uy,
A(Uy) =U(y).
, ,
AU=U = U1AU=
A= UU1.
:
5.3. A M(n) , vi
U= (v1, v2, . . . , vn)
A= UU1 U1AU= (5.6) .
:
i vi, :
Avi=ivi.
n , :
A(v1 v2 . . . vn) = (1v1 2v2 . . . nvn).
AU=U,
! ,
.
5.2. .
,
, , -
.
. Jordan.
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74 5.
5.5 -
: , , - A I . det(AI) n, - A. A M(n; R) n, .
: i, AiI- i. k(k 1 ), - , ,
k.
(A iI)v= 0.
, -
. - k = 1( ), cvi, vi .
: A M(n) -, ,
U = (v1 vn)(vi) A- .
A= UU1 U1AU=
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5.6 75
5.6 1. :
A=
1 32 0
, B=
3 21 2
, C=
5 50 2
.
2. :
A= 1 4
2 4 , B=
3 3
1 1 .
3. :
A=
2 3 10 1 0
0 0 2
, B= 2 1 21 2 0
2 0 0
,
C=
1 1 22 0 2
1 1 3
, D=
2 0 20 2 0
2 0 2
.
: C = 2.
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76 5.
5/21/2018 . -
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I
5/21/2018 . -
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5/21/2018 . -
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:
, .
M(n) , A, det A .
.
.1. A =
a bc d
.
A
det A= ad bc. det A =|A|. -
, A( ) , .
.1. v1, v2 R2
A= v1 v2 .
. ,
, .
, v1, v2 (1, 2) = (0, 0)
1v1+ 2v2 = 0.
(
, )
5/21/2018 . -
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80 . :
2 , . v2=v1, = 1/2. :
(c, d) =(a, b) = det A= (ab ba) = 0., , ad = bc,
a/b = c/d, ( b, d = 0;)
M(3): , : A
A=
a11 a12 a13a21 a22 a23
a31 a32 a33
2 2
det A= a11
a22 a23a32 a33 a12 a21 a33a23 a31+ a13 a21 a32a22 a31 . (.1)
:
: ,
a13( ) A . .
det A= a11a22a33+ a12a23a31+ a21a32a13 a31a22a13 a21a12a33 a32a23a11,(.2)
-
450 45o.
.1. .
, (.1),
M(n),n 4. .2. A M(n) n. (i, j), Aij(n 1) (n 1) A i j. (i, j) A (principal minor). (i, j) ij A(cofactor)
ij = (1)i+j|Aij|.
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81
.1. (1)i+j , .
.3. A M(n) det A= |A| =a1111+ a1212+ . . .+ a1n1n.
,
(
.1.)
.1. a1, a2, . . . , an Rn ( ) A =
a1 a2 . . . an
.
( .)
.2. 1. A M(n)
det A= ai1i1+ ai2i2+ . . .+ ainin
i = 1, 2, . . . , n. i -. j:
det A= a1j1j+ a2j2j+ . . .+ anjnj
2. i =k,ai1k1+ ai2k2+ . . .+ ainkn = 0,
i k, . .
-
, .2:
.2( ). 1. A,AT, :
det AT = det A.
2. A, . .
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82 . :
3. ( ) A , . , An ,
det(A) =n det A.
4. ( ) A ( ), (-
,
( .)
5. det A
ai ak, ai ai+ak. .
6. ( )
.
( ,
) :
.3. A, B M(n), det AB= det A det B.
- : m= n, - A M(m, n). , - A, min(m, n)! - .
.4. k min(m, n) k , 1 i1 < i2