, .
) .
, = + , , .
, L , ,
,
(*).
(*) , , (, ) = (, ) = (, ) = 0, :
:
, :
; ; .
,
, L: , : .
) - L, :
) ³
VI. , R.
. A- . , . , .
A (X+Y)
A X
A Y
A X+A Y
, A (X+Y)= A X+A Y.
- . A ( X) (A X)
: A .
.
䳺 A.
A
A
A
A
A
VII. A , B . A B , .
. A e, , , B . , T1 T2 a i b e . , a i b , a i b, ,
:
³: .
VIII. KerA, ImA , A - L3, .
. A , r(A): . r(A)=2, dim(ImA)=2, A 22. dim(ImA)+dim(KerA)=n, dim(KerA)=32=1, 1. KerA ImA .
ImA = < xA R 3 >, , , , e 1A, e 2A, e 3A.
:
e 1A =(1,2,3); e 2A =(2,1,1); e 3A =(1,0,1).
r(A)=2, , . e 1A e 2A; , ImA e 1A =(1,2,3); e 2A =(2,1,1). ImA .
KerA. KerA , KerA R 3, [x]A=[ ],
|
|
, , 2. 1. ,
Ker A .
³: ImA . KerA . dim(ImA)=2; dim(KerA)=1.
IX. A, B=<b1,b2,b3>
.
,
,
, :
, .
:
r=2;
, ,
, L1, (1,0,1).
.
r=2
x=(0,1,0), ,
, , , (0,1,0).
:
r=2.
.
, , :
, L2, (-2,3,0).
, , , , L1,L2,L3, (1,0,1), (0,1,0), (-2,3,0).
. A V3 ? .
. n n ( ), .
:
,
.
, 1.
, 1:
³:
4
. f :
) ;
) f ;
) f ;
) , f
;
) , f .
1. f=x12+5x22+4x32-2x1x2+4x1x3;
2. f=4x12+4x22+x32-4x1x2+4x1x3-3x2x3;
3. f=x12-3x32+2x1x2-6x2x3;
4. f=7x12+6x22+5x32-4x1x2-4x2x3;
5. f=6x12-2x22+6x32+4x1x3;
6. f=2x12+5x22+x32+2x1x2+2x2x3-16x1x3;
7. f=x12-2x22+3x32+4x1x2-4x2x3-8x1x3;
8. f=2x1x2+3x2x3-x1x3;
9. f=5x1x2-x2x3+x1x3;
10. f=4x12+5x22+6x32-4x1x2+4x2x3.
II. ,
1. A= ; 2. A= 3. A=
|
|
4. A= 5. A= 6. A=
7. A= ; 8. A= 9. A=
10. A=
, 3 . .
1. f=2x1x2+x22+x12+x3 2 +4x1x3+2x2x3;
2. f=11x12+5x22+2x32+16x1x2+4x1x3-20x2x3;
3. f=x12+x22+5x32-6x1x2+6x1x3 6x2x3;
4. f=x12+x22+x32+4x1x2+4x1 x3+42x3;
5. f=17x12+14x22+14x32 4x1 x2-4x 1x3-8x2 x3;
6. f=6x12+5x22+7x32-4x1x2+4x1x3;
7. f=4x12+x22+x32-4x1x2+4x 1x3-3x2x3;
8. f=x12+2x22+3x32-4x1x2-4x2x3;
9. f=2x12+x22-4x1x2-4x2x3;
10. f=5x12+7x22+6x32-4x1x3+4x2x3.
IV. , .
1. x12+y2+z32+2xy+4xz+2yz-6x+8y-2z-5=0;
2. 11x2+5y2+2z2+16xy+4xz-20yz+4x-6y+8z+1=0;
3. x12+y2+5z2-6xy+6xz-6yz-2x+4y-6z-4=0;
4. x2+y2+z2+4xy+4xz+4yz-6x+4y-2z-1=0;
5. 17x2+14y2+14z2-4xy-4xz-8yz-2x+6y-8z-4=0;
6. 6x2+5y2+7z2-4xy+4xz-8x+2y-2z+3=0;
7. 4x2+y2+z2-4xy+4xz-3yz+4y-6z+2y-4=0;
8. x2+2y2+3z2-4xy-4yz+2y+4z-2x+1=0;
9. 2x2+y2-4xy-4yz+2x-2y+6z-2=0;
10. 5x2+7y2+6z2-4xz+4yz-4x-2y+8z-2=0;
4
. f :
) ;
) ;
) f ;
) , f
;
) , f .
:
f=2x1x2-12x1x3-x22-8x32;
) f:
A= =
.
A= ~ ~ r=3.
) f :
= , XT=(1,2,3) f=XTAX
ij, =(1,2,3) .
) f .
1- :
1- 2.
Q= .
, Q
Q/E= ~
, G-1= , (Q-1)T=
f1, f Q:
= TAf Q-1
= = .
f1:
f1=y12-y22-8y32-12y1y3.
2- :
,
1. .
.
.
= .
f2:
=z12-z22-44z32, .
f2 : , .
= .
, f3=t12-t22-t32 f.
,