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, . , . .

, , . ࠫ.頫黠 , ().. , . . (, , ), . , , . .,, . . ࠫ . ,. ܠ 1. Def: V () K, V *) : , ( +), K, .), ( :

. x, y V zV z= x y 1) x y = y x ;2) ( x y) z = x ( y z) ;

3) V x = x ;4) x V y V x y =.

. x V, zV z = 1)1K;1x=x2) x V,, (x)=()x.

, *) .

.:

( + ) x = x x;

, x V 1) x, y V ( x y) = x y 2) K R ( ), V . K ( ), V .

() .

W V 1) x, y W x y W, 2) x W x W, W V.

x = 1e1 + 2e2 + + k ek ( i, ei V, i = 1, 2,, k) 1,2,,k.

{ ei } i=1 k ( e1,e2,,ek ).

( e1,e2,,ek ) = V,{ ei } i=1 V.

{ ei } i=1 , i ei = 0,1 = 2 = = k = 0.

V { ei } i=1 V,:

)V )V )V.

VdimV.

V{ ei } i=1, x V 1, 2,, n , x = 1e1 + 2e2 + + nen. { e1,e2,,en}, 1, 2,, n { ei } i=1.

V V , ( x x),,: x x y y, x y x y x x'. V V , dimV=dimV.

L1L2V.LN :



.

: dimL1 + L2 ) = dim 1 + dim 2 dimL1 L2 ).

L = L1 + L2 ( L1 L2 ), x = x1 x2 ( x1 L1, x2 L2 ) x L.

蠠LV蠠0V,̠ {x|x=0y, yL} V. L ,0V.

0L ( ) ̠ VL .

Def: V K ( ) ,:

n m: aij (). :()ij.

Def:,頠ij=aji,ij =aji().

.

:

K:()ij=(ij);

:(+)ij=()ij+()ij;

n x m. m x k.( 1 2 ):

:;

( ( ,*+): ( A+ ) ij = ( A ) ij = ( A) ji.

,1),2)3) (..nxn)n.

1. .

?.

) ;) ?.

4. .

VV? ,, , , V.

.

V:)2;)1,2,,n,?.

9. .

10. 11.?.

12..

n? ?

15.?

, ,?

, .

1. R+ ( ), :1),R+xyx.y;2)R,R+ xx.

. ,.

., .:

.1)xyx.y=y.xyx.

2)(xy z)(x.y).z=x.(y.z)x (y z).

3) =1xx.1=x.

4)R+ yR+=1/|xyx.1/1.

.1)1R11=.

2) ( )( ) = (.) .

.1)(+) 2) ( )(. ) = . .

. + R. , , R+ .,R+ . 젠0(0 1). R+= x ,1 1. log0{0}.

2. m n 1,2,,n , Vn.

. (1,2,, n). : aik xk = ( x1, x1,, x1 ) ( x1, x2,, xn ) . ( x1 + x1, x1 + x2,, x1 + xn ) aik ( x1 + xk ) = aikx1 + aikxk = 0. , . ( x1, x2,, xn ) . (x,x,,x ) = aik xk = 0,..,, . Vn (1,2,,n).

3. :(3)=(2)=0.

. , = 3 = 2 ,4():

4()=(2)3()=(2)(3)2()=(2)(3)(0+1+22)= =0(2)(3)+1(2)(3)+22(2)(3). , :{(2)(3),(2)(3),2(2)( 3)}. , {(2)(3), (2)(3), 2(2)( 3)} . : (0+1+22)(2)(3)=..(2)(3)(0+1+22)0. , 蠠=0 0=0;

蠠=1 1+2=0;

蠠=1 1+2= 0=1=2=0. ,{(2)( 3), ( 2)( 3), 2( 2)( 3)} .

, 4 , 3 {(2)(3),(2)(3),2(2)(3)}.

4.:1(1,2,3,4), 2(0,1.0,1),3(1,1,1,1),4(4,1,6,3)?.

. 11 + 22 + 33 + 44 = . . , . ,:

31+3+64=21+(1+3+44)+24=01=44+3+ +44=03=34;44+2+34+34=02=24. :41242343+44=0 .4=1,4=1+22+33.

.

5.,cosx,cos2x,cos3x(x[0,2]) [0,2]. . , . , , .. : cosx+cos2x+ cos3x 0, 砠, . ࠠ,ࠠ,젠:cosx +4cos2x +9 cos3x 0, cosx +16cos2x +81 cos3x 0. = 0, , = = = 0. ,, .,.

6.(1,2,3), 堠1(1,2,3),2(3,0,1),3(1,10,17).

.1)1,2,3 ꠠ:1+2+ 3=;(1,2,3)+(3,0,1)+(1,10,17)=(0,0,0) 2 , = 5; 堠1,:5+3=0..=2; : 51 +32 3 =0.. 3 =51 22. 3 1 2, {1, 2, 3} {1, 2} .

2) 1+2=(1,2,3)+(3,0,1)=,.. , , , 1 2 . dim(a1,a2,a3)=2,,,{1,2}.

7. .: , :

,,:

,, ., 3:

8.(4,7,4){a1,a2,a3}, a1(2,3,1),a2(1,1,0),a3(0,2,1).

.=a1+a2+ a3., ,:

젠=1+22+33.,⠠{a1, a2,a3}(1,2,3).

9.蠠񠠠蠠L1+L2蠠L1L2,L1 = {a1(2,1,1), a2(3,5,4), a3(1,3,6)}, L2 = {b1(1,1,2), b2(6,5,1), b3(7,4,1)}. . 1) , {a1, a2, a3} .砠ࠠa1+a2+ a3=, ࠠ=2,=1, =1.,2a1+a2+a3=a3=2a1a2. ,{a1,a2,a3}={a1,a2}dim{a1,a2}=2.

2),{b1,b2,b3}. ࠠb1+b2+ b3=, ࠠ=1,=3, =1.,b1+3b2+b3=b3=b13b2. ,{b1,b2,b3}={b1,b2}dim{b1,b2}=2.

3)堠L1L2.蠠L1L2 =1+2= b1+b2. ,=,=2,=.21+2= =b1+b2= .(3,5,1).,L1L2={3,5,1}. ࠠdimL1L2=1;dim(L1+L2)=dimL1+dimL2dimL1L2=3. 젠L1+L2,,⠠{1,2,b1}.

4. 1., n .

2. n?

.,3 ( 2x 3 + 5x 2 + 3x 5) ( 2x 3 + 5x 2 7x + 12) 2蠠 3. , ,( )?

.,ࠠn=dimV xV,x ,xP Q;

)dimP+dimQ=dimP Q+1, .

66. , P V, Q,V=P Q.

67. {ai}{bi}:

)1(1,2,0,1),2(1,1,1,0);b1(1,0,1,0),b2(1,1,1,1); )1(1,1,1,1),2(1,1,1,1),3(1,3,1,3);b1(1,2,0,2),b2(1,2,1,2), b3(3,1,3,1). )3;1;)3;2. 68. {ai}{bi}:

)1(1,2,1),2(1,1,1),3(1,3,3);b1(2,3,1),b2(1,2,2),b3(1,1,3);

)1(1,1,0,0),2(0,1,1,0),3(0,0,1,1);b1(1,1,1,1),b2(1,0,1,1), b3(1,3,0,4);

)1(1,2,1,2),2(2,3,1,0),3(1,2,2,3);b1(1,1,1,1),b2(1,0,1,1), b3(1,3,0,4);

)1(1,1,0,0),2(0,1,0,1),3(0,0,1,1);b1(1,0,1,0),b2(0,2,1,1), b3(1,2,1,2).

){1,2,b1};{21+2=b1+b2=(3,5,1)};){1,2,3,b2};{1+3=b1=(1,1,1,1)}; ){1,2,3,b2};{b1=21+2+3;b3=5123};){1,2,3,b2};{b1=12+3= =(1,0,1,0);21+23=b1+b3=(2,2,2,2)}. 69. 3, :

1+2t+t3,1+t+t2,tt2+t3蠠1+t2,1+3t+t3,3tt2+t3.

:3;:{1+2t+t3,1+t2,1+t+t2};:1;:{2+3t+t2+t3}. 70. ) , n ,2n;

) . ࠠ(3 +2i,i). ):{(1,0),(i,0),(0,1),(0,i)};(3+2i,i)=(3,2,0,1). 71. n , ,⠠{ai}{bj}:

)n=3;1(1,2,3),2(1,2,i),3(2,0,3+i);b1(1,0,3i),b2(1,4,3+i), b3(1,4,34i); )n=3;1(1,i,1+i),2(1,0,3i),3(1,2i,2+i);b1(1,2,i), b2(2,1+i,i),b3(1,4,34i); )n=4;1(1,1,1,1),2(1,2,1,3i),3(2,3,2,4i),4(1,1,1,1i); b1(0,1,0,3i),b2(0,2,0,52i,i),b3(0,2+i,0,6+i),b3(1,4+i,5i,2i).

a)3;:1,2,b1;1;:(0,4,3i);)3;:1,2,b1;1;:(9+10i,216i,103i);

)4;:1,2,2,b4;2;:b1,b2. 72. , n .  :

);

)1 2i + ( 3 + i ) t 3t2 堠 (n=2).

a):dimV=n+1,:1,t,t2,tn;: dimV=2n+2,:1,i,t,it,tn,itn;):(12i,3+i,3);

:(1,2,3,1,3,0). 73. +.,:

a) 3 11 ;) 3i 74.:

75. , A+ = A ;, A+ A = E.

, AT = A ; , AT A = E. :

a););););

);););). 76.n,, V6: )nn ;

)nn;

) () n, :(0)=0. a)n=4;)n=5;)n=6.

1. ....:,1975.

2. ....:,1971.

3. ....:,1970.

4. ..,....:,1973.

.. . ..:,1969.

6. .. . .: ,1985.

.., .., .. ..:,2001.

8. .. . .:, V K , (K C) .

(K R) . ( ,).

V { ei}n, i j .

() ࠖ=   0  .

,(,)=  ., V f1,f2,,fn : +1e1+2e2蠠1,2頠e1e3, ek , . , . V V , ,,x,yV x,y V ,xx,yy. ࠠ(x,y)=(x,y ).

, .

hL(hL)蠠yL(h,y)=0.

젠L L:

,L 젠:

)V=LL,)dimL+dimL =dimV. LV:

x0xL, x 頠頠ࠠxL.

x (|x |)젠x젠L.

Vx x0 xL.

{ ei } 1n V, ̠ = + L L, L L. .

.

[,b]..

() ?

?

?

, , .?

? .

().

n?

.

()?

L?

, L L .

LL ?

L, L?

L?

1. , ,.

. , . 13,,ࠠ (,)=(,)=(,)=2(,).

⠠=i,(i,i)=(,)0. ,(f,f)=0,f()=0,..f=.(f,f)=0,..f2(1)+ f2(0) + f2(1) = 0. f( 1) = f(0) = = f(1) = 0. f() 2,, .,f()=0.

,,, 2.

 3,f(x)=x(x2 1)(f,f)=0,f ,4.

4.2,.

)f()=1+2蠠g()=1+.

) 2 堠0=1,1=,2=2.

.)f()g()=1,=0,=1:f(1)=3,f(0)=1,f(1)=1, g(1)=0,g(0)=1,g(1)=2.(f,g)=f(1).g(1)+f(0).g(0)+f(1). g(1) (f,g)=3.0+1.1+1.2=3,(f,f)=32+12+12=11,(g,g)=02+12+22=5.

fg:

) f(x)g(x):0 =1, 1 =, 2=2. f(x)=0+1+22,g(x)=d0+d1x+d2x2; f = ci pi, g = cj pj f g : ( f, g) = ijci dj,堠 ij =(pi,pj). ij , ., . ij: 00=(0,p0)=3; 11=(1,p1)=2; 22=(2,p2)=2; 01= 10=(0,p1)=0; 02= 20=(0,p2)=2; 12= 21=(1,p2)=0.

: = 0 2 0, f(x) g(x) 0=1,1=,2=2:

(f,g)=30d0+21d1+22d2+20d2+22d0.

5. 22(f,g)=f( 1).g(1)+f(0).g(0)+f(1).g(1). . 2 , , 1 = 1, 2 = , 3 = 2, 1,2,3,. 젠=(2,1).(1,1)1,=(3,1).(1,1)1, =(3,2).(2,2)1. :

,=02=2=.,=2/3.,(3,2)=(1)2.(1)+ 02.0+12.1=0,, =03=3(2/3)1=2(2/3).

,: 1,2,3(..), 1,2,3.ꠠ(1,1)=3,(2,2)=2,(3,3)=2/3, 6. [0,] : f1(x)=1,f2(x)=x,f3(x)=x2.

.{f1,f2,f3}:1=f1()=1, 2=f1+1.1,(1,2)=0, ,:

2=(/2).

3:3=f3+1+2.⠠蠠:

:3=f3((24)/2)12=2+2.

{1,2,3}., ⠠1,2蠠3:

,{f1,f2,f3}: 7. 22젠 ( f, g) = f ( x ) g( x ) sinxdxf=x +x+1堠 = 1,2=(/2),3=2+2.

..f:f=1+2 + 3.,, f{1,2,3}.

1 (f,1)=(1,1)+(2,1)+ (3,1),,..,(2,1)=(3, 1)=0. , f堠{1,2,3}: .f=x2+x+1:

x2+x+1=(x2x+2)+x2+x+1=(x2x+2)+(+1)1= =(x2x+2)+(+1)(x )+ ,堲.

8.,:1,sinx,cosx,sin2x,cos2x,,sinnx,cosnx , : .

.젠,:0=1,fk=sinkx,gm=cosmx(k,m=1, 2,3,,n).

:

( e0, fk ) = sinkxdx;2) ( e0, gk ) = coskxdx;

4)5)k l . :

,,, 9.1,2,3,4:1(1,0, 1,1,2),2(1,0,1,1,2),3(1,0,3,0,0),4(0,0,2,1,6). )蠠(1,2,3,4); )L ꠠ(1,2,3,4).

.)(1,2,3,4). ,, ,4=122+3,..41,2,3. ,(1,2,3,4)=(1,2,3).⠠1,2, 3,1+2+ 3=0.,,: = = = 0. 1,2,3(1,2,3,4).

젠{1,2,3}:1=1=(1,0.1,1,2),2 ,2 ,2: e3 = x3 e1 e2 = (1 0, 5, 4, 0).Ƞ堠3:

1=(1,0,1,1,2),2=(2,0,2,2,3)蠠3=(1,0,5,4,0) (1,2,3,4).

) (1,2,3,4)= (1,2,3).蠠 (1,2,3,4, 4) 12,,,5=0.,2 ,2.1,3,4:, ,1=33,4=23.

,2蠠3,2=1,3=0蠠2=0,3=1 ,:4(0,1,0,0,0)蠠5(3,0,1,2,0).45 (1,2,3),,..{3,4} (1, 2,3).

10. (1,1,1,1,1)蠠 1(1,0,1,1,2), 2(2,0,2,2,3),3(1,0,5,4,0) .

)z ࠠ󠠠(1,2,3):=+z,堠L== (1,2,3),zL ;

)(1,2,3);

)ࠠ(1,2,3).

.)젠=+z,堠L,zL. L,=1+2+ 3=1+2+ 3+z.

1 , zL, .. (z, 1) = 0 {1, 2, 3} ,..(1,2)=(1,3)=0.: = ( e2,e2 ) 21, ( e3,e3 ) 21. y = 1(1,0,1,1,2) 2(2,0,2,2,3)+ 3(1,0,5,4,0)= = (1,0,9,3,7). 堠z==(1,1,1,1,1) (1,0,9,3,7)== (6,7,2,4,0). , (1, 2, 3) 𠠠= (1,0,9,3,7),頠頖z == (6,7,2,4,0).

) (1, 2, 3) 젠頠ࠠ(1,2,3). )젠蠠(1,2,3) z, .. d = z = ( z,z) =. , z頠, 4) , :

)?

)?

)?

););). 5) (.) (1,2)蠠(1,2):

)(,)=21;)(,)=11+212+1022;

)(,)=12+21;)(,)=211+322;

)(,)=11212221+522;

)(,)=711+612+621+922;

)(,)=911312321+22;

)(,)=211+212+221+22.

),);10;);40;),),););40;). 3., F(x, y)= a11x1y1 + a12x1y2 + a21x2y1 + a22x2y2, x1, x2, y1, y2 x y , ,ࠠa21=a12,a11>0,a11a22 a12 >0.

(x,y) x(1,x2)蠠y(y1,y2):

)(,)=11+22;)(,)= )(,)=i )(,)=(3+i)1 y 2 +(3i)2 y1 ;)(,)=31 y1 +42 y 2 ; )(,)=5 ),);10;);10;);40;);40; ););10;). 5. 砠1,2蠠1,2⠠蠠 ,a11,a12, a21蠠a22堠蠠F(x,y)==a11x a21x2 y1 +a22x2 y 2 .

a21= a12;a11>0;a22>0;a11a22|a12|2>0. 6.  :

?

n=3,;40;n=2,. 7.   n ););40;). 8. , n :

PQ; PQ=,ࠠR.

9. , :

f ( x ) g( x )xdxࠠ(1,1);) f ( x ) g( x )xdxࠠ(0,1);

f ( x ) g( x )x dx ࠠ(1,1);) f ( x ) g( x )dxࠠ(1,1); f ( x ) g ( x ) xdxࠠ(0,1);) f ( ) g( ) sin d ࠠ(0,); f ( x ) g( x )dxࠠ(0,1);) f ( x ) g( x ) dxࠠ(0,1);

);40;);););40;);2030; ););),V,V;). 10. )(5,4,3),;

)(1,2,3,4),; )(1,1),(,)=11+12+21+322;

)(1,1,2),(,)=411+212+221+2222332+333;

)(1,i),(,)=2 )(1+i,1i),(,)= +(2i) .

11. 3:

)(2,3,1),(4,6,2);)(1,1,2),(1,0,1);)(1,0,1),(1,2,2). )0;)/6;)3/4. 12. 4:

)(1,1,1,1),(1,1,1,1);)(1,2,3,4),(5,0,2,1); )(1,2,2,1),(1,1,1,2). );)2/3;)arccos 7 10. 13. 2 (, )= 11 + 12 + 21 + )(1,0),(0,1);)(1,0),(1,1). )arccos 1 / 3 ;)/2. 14. 3젠(,)=411+212+221+ 2222332+333:

)(1,0,0),(0,1,0);)(1,0,0),(1,2,2). )/4;)/2. 15. 4젠 (,)蠠:

)(2,1,3,1),(1,2,2,2);)(1,2,2,3),(3,1,5,1);

)(1,1,1,2),(3,1,1,0). )0,/2;)18,/4;)3,arccos 3 / 77. 16. , (1,1,1,1),(1,1,1,1),(2,1,1,3).

⠠ 0,1 / 2, 1 / 2, 0. 17. , :

)(2,2,1),(2,2,3);

)(1,2,1,0),(1,1,1,1); )(1,1,2),; )(1,1),(,)=11+12+21+322;

)(1,2,0),(2,0,1),(,)=411+212+221+2222332+333;

)(1+i,1i),;

)(1,1+i,0),.

)1/3(1,2,2);),1/2(1,1,1,1);), 1 / 2 (1,1,0);) 1 / 3 (2,1); ) 1 / 7 (1,1,1);)1/2(1+i,1+i);)1/2(1i,1,i). 18. )(1,3,1),(4,5,3);)(1,0,1,3),(2,1,1,3);

)(2,0,1),(4,5,3),(1,7,3);)(1,2,2),(1,1,0),(0,1,4);

)(2,1,2),(4,1,0),(0,1,0);)(1,1,1,0),(1,2,0,1),(0,0,1,0).

18. )(1,3,1),(2,1,1);)(1,0,1,2),(1,2,1,0);)(2,0,1),(1,1,2),(1,5,2); )(1,2,2),(2,1,2),(2,2,1);)(2,1,2),(1,0,1),(1,4,1); )(1,1,1,0),(0,1,1,1),(1,0,1,1). 19. . , :

)(1,2,1),(4,4,1),(1,3,1);)(1,1,1),(4,0,5),(8,2,0);

)(3,1,2),(4,0,1),(5,1,0);)(1,2,1,1),(5,5,4,2),(3,6,2,0);

)(1,0,1,1),(6,0,4,5),(3,2,5,4); )(1,3,2,1),(1,7,3,2),(2,2,3,1);

)(1,1,1,1),(4,2,4,2),(2,7,4,7),(2,7,2,5);

)(i,1,i),(2,0,1),(0,2,i);

)(1,1,1+i),(1,i,2+i),(1+2i,4i,1).

) 1/ 6 (1,2,1), 1 / 2 (1,0,1), 1 / 3 (1,1,1);) 1 / 3 (1,1,1), 1/ 14(1,3,2), 1/ 42(5,1,4);) 1/ 14(3,1,2), 1 / 3 (1,1,1);) 1/ 7 (1,2,1,1), 1/ 7 (2,1,1,1), 1/ 14(0,2,1,3);) 1 / 3 (1,0,1,1), 1 / 2 (1,0,1,0), 1/ 10(1,2,1,2);) 1/ 15(1,3,2,1), 1 / 3 (1,1,1,0);)1/2(1,1,1,1),1/2(1,1,1,1), 1 / 2 (1,0,1,0), 1 / 2 (0,1,0,1); ) 1 / 3 (i,1,i), 1 / 2 (1,i,0);)1/2(1,1,1+i), 1 / 3 (1i,0,1), 3 / 6 (1,3,1+i). 20. f1, f2, f3 1,2,3,: )f1(1,0,0),f2(0,1,0),f3(0,0,1);

)(,)=411+212+221+2222332+333;

)(,)=11+212+221+522223232+733;

)f1(1,0,2),f2(0,1,2),f3(4,0,0);

(,)=411+212+221+2222332+333;

)f1(0,1,1),f2(2,0,1),f3(2,4,0);

(,)=5111221+22+213+231+433.

.)(1/2,0,0),(1/2,1,0), 1/ 2 2 (1,2,2);.)(1,0,0),(2,1,0), 1 / 3 (4,2,1), )1/4(1,0,2), 1 / 2 (1,1,0),1/4(1,4,2),) 1/ 5 (0,1,1), 1/ 2 3 (2,2,1), 1/ 2 5 (0,4,1). 21. 1, 2, 3 .,: 1 =1/3(21+223); 2 =1/3(212+23); 1 = 1/3(122 23) 22. :ij,i+2j,i+jk.

1 / 2 (ij), 1 / 2 (i+j),k. 23. , , :

)(1,2,1,3),(4,1,1,1),(3,1,1,0); )(1,2,2,1),(1,1,5,3),(3,2,8,7);

)(2,1,3,1),(7,4,3,3),(1,1,6,0),(5.7,7,8).

)(1,2,1,3),(10,1,1,3),(19,87,61,72);)(1,2,2,1),(2,3,3,2),(2,1,1,2); )(2,1,3,1),(3,2,3,1),(1,5,1,10). 24.  V4:

)(1,1,1,2),(1,0,1,1);)(1,1,2,0),(1,1,1,3);

)(1,2,1,2),(1,1,1,1);)(4,0,1,0),(1,6,1,2). )(1,0,1,0),(1,6,1,2);)(1,1,0,0),(1,1,1,1);)(1,1,1,1),(2,1,2,1); )(1,1,1,2),(1,0,1,1). 25., 1(1/2,1/2,1/2,1/2),2(1/6,1/6,1/2,5/6).

1,2,3= 1 / 26 (0,4,3,1),4= 1 / 3 26 (13,5,6,2). 3) , :

)(1,2,2,3),(2,3,2,4);)(1,1,1,2),(1,2,3,3).

)(2,2,1,0),(5,2,6,1);)(1,2,1,0),(25,4,17,6). 4) , :

)(2/3,1/3,2/3),(1/3,2/3,2/3);)(1/2,1/2,1/2,1/2),(1/2,1/2,1/2,1/2).

)(2/3,2/3,1/3);)(1/2,1/2,1/2,1/2),(1/2,1/2,1/2,1/2). 5) 2 (0, 1) 3. , 1,,2,1, ( f, g ) = f ( x ) g ( 2 ) dx.I.

7) ,:

. , L0, L1, , Ln n n 8)   31,, 2, 3 . ,,:

1, , 2, 3 , ); )I;) ࠠ.

9) V (0,+ ), 堠 {,x,x2x,}., ,,蠠 .

10) L L,: 1(1,0,2,1),2(0,1,2,1).

,b1(2,2,1,0),b2(1,1,0,1). 11) :(ij). (i+j)+ k. 12) L L, ,(1,0,6,0),(3,2,0,2). 13) 3 , = 0. ) ;) ;)33+22+1+0.

)(0,3/5,0,1);)(1/3,0,1,0);)(0+2/3,1+(3/5)3,2,3). 15) 3 +1(1,), :

)1(2/5);)(2/7). 16) 3i+1 (1,) 17) L . L , )(10,20,10),(0,1,0);)1(1,1,1),2(4,0,5),(7,3,1); )(4,3,2,1),(1,1,1,1);)1(1,1,1,0),2(2,1,0,1),(1,0,2,2);

)1(1,1,1,1),2(1,4,1,0),(2,1,1,0);

)1(1,0,1,1),2(3,3,2,1),3(1,6,3,5),(1,4,0,2);

)1(2,0,1,1),2(1,1,1,1),3(1,1,1,1),(1,2,0,1);

)1(1,1,1,1),2(5,1,1,3),3(3,1,1,0),(5,4,3,2).

)1/3(1,2,1),1/3(1,1,1);)(2,2,3),(5,1,4);)1/15(4,3,2,1),1/15(11,18,13, )(0,1,2,1),(1,1,0,1);)1/2(3,2,1,2),1/2(1,0,1,2);)(2,3,1,0),(1,1,1,2); )1/2(1,3,1,3),1/2(1,1,1,1);)(3,0,1,2),(2,4,2,4). 18) z(1,2,,k):

)(4,1,3,4),1(1,1,1,1),2(1,2,2,1),3(1,0,0,3); )(5,2,2,2),1(2,1,1,1),2(1,1,3,0),3(1,2,8,1); )(7,4,1,2),L: 3 x1 + 2 x2 + 2 x3 + x4 = 0.

)=3122=(1,1,1,5);z=(3,0,2,1);)=212=(3,1,1,2);z=(2,1,1,4); )=(5,5,2,1);z=(2,1,1,3). 19) 蠠 zࠠࠠL堠1,2,,k:

)(5,2,2,2),1(2,1,1,1),2(1,1,3,0); )(3,5,9,3),1(1,1,1,1),2(2,1,1,1),3(2,7,1,1). )(3,1,1,2),z(2,1,1,4);)(1,7,3,3),z(4,2,6,0). 20) L )(0,1,1),f1(1,0,1),f2(0,1,0);)(5,1,2),f1(1,1,2),f2(1,1,3); )(1,1,0),f1(0,2,1),f2(1,3,0);)(1,1,2),f1(1,2,1),f2(2,1,1);

)(2,2,1,0),f1(1,2,3,0),f2(1,0,1,2); )(1,0,2,1),f1(3,3,1,1),f2(2,1,0,2); )(1,1,0,1),f1(1,1,1,1),f2(1,1,3,3),f3(3,2,4,1); )(1,1,1,3),f1(1,1,1,2),f2(1,0,1,1),f3(3,1,2,1); )/6;)/2;)arccos 3 7 ;)0;)arccos 1 / 3 ;)/2;)/4;)/6. 21) (1,2,,k)젠:

)(1,3,1,3),1(1,1,1,1),2(5,1,3,3); )(2,2,1,1),1(1,1,1,1),2(1,2,3,1),3(1,0,5,3);

)(2,2,1,1),1(3,4,4,1),2(0,1,1,2).

)/4;)/2;)/3. 22) 0+11+22++mm:

)(1,2,1,1),0(0,1,1,1),1(0,3,1,5),2(4,1,3,3);

)(0,0,0,0),0(1,1,1,1),1(1,2,3,4).

) 7 ;) 2 3. 23) , ,:

)(4,2,5,1);)(4,1,3,7); )5;)7. 24) ,n. :n+1n1+2n2++n1+n.

25) , 젠1,2,,k, .. 1, 2, , k b1,b2,,bkG(1,2,,k)=G(b1,b2,, bk) = (b1, b1)(b2,b2)(bkbk). 19.G(1,2)=1,2; G(1,2,3)=1,2,3. 26) g1,g2,, gn.,1,2,,ng1,g2,, gn: 27) (. . 48), : )(1,2,1),(4,4,1),(1,3,1); )(1,1,1),(4,0,5),(8,2,0);

)(3,1,2),(4,0,1),(5,1,0). ) 1/ 6 (1,2,1), 1 / 2 (1,0,1), 1 / 3 (1,1,1);) 1 / 3 (1,1,1), 1/ 14(1,3,2), 1/ 42(5,1,4);) 1/ 14(3,1,2), 1 / 3 (1,1,1).

9. ....:,1975.

10. ....:,1971.

11. ....:,1970.

12. ..,....:,1973.

13. .....:,1969.

14. ....:,1985.

15. .., .., .. . .: ,2001.

16... . .:, ܠ

ߠȠߠ

Def: naij, i , j j ,.. (det A) j (1, 2, , n) : )(): (1,,i1+i2,,n)=(1,,i1,,n)+(1,,i2,,n);

) ( ):(1,,i,,j,,n)=(1,,j,,i,,n);

): ( e1,e2,,en ) = = 1.

, ,:

N(j1j2jn) j j j j ,,蠠jk>jm蠠km.

35. , , 0, .

36.,, ,.

37., :

) ;) ;) )1=1,(1,1);2=3,(1,1); 0;)1=7,(1,1);2=2,(4,5); 0; )1=ai,(1,i);2=ai,(1,i); 0;)=2,1(2,1,0)+2(1,0,1); c1 + c2 0 ; )1=1,(1,1,1);2=1,(2,0,1);3=2,(1,0,0); 0;)1=1,1(1,0,1)+2(0,1, 2=1,(1,0,1); 0; c1 + c2 0 ;)1=1,(3,6,20);2=2,(0,0,1); 0; )1=2,(1,0,0);2=1,(1,0,1);3=1,(0,1,1); 0;)1=1,(1,0,0);2=1, 1(1,0,1)+2(0,1,1); 0; c1 + c2 0 ;)=2,(1,0,0);)=2,1(1,2,0)+2(0,0,1); c1 + c2 0 ;)1=1,(1,1,1);2=0,(1,2,3); 0. 38. f(x) =;f(x)=cex,c 0. 39., :

14. s1,s2,s3:1=1,2=0,3=1; :ys1:c1(0,i,1),c2(1,0,0),c3(0,i,1),ci 0; ys2:c1(i,0,1),c2(0,1,0),c3(i,0,1),ci 0; ys3:c1(i,1,0),c2(0,0,1),c3(i,1,0),ci 0. 40., ,:

15. ⠠s1,s2,s3:1=1,2=0,3=1; :s1:c1(1, 2,1),c2(1,0,1),c3(0, 2,1),ci 0; s2:c1(1,i 2,1),c2(1,0,1),c3(1,i 2,1),ci 0; s3:c1(1,0,0),c2(0,1,0),c3(0,0,1),ci 0. 41. f(x). f():

)1=6(1i),(1,i);2=6(1i),(1,i); 0;)1=14,(1,2,2);2=3=10, (1,2,1); 0;)1=9,(1,0,0);2=1,(1,0,1);3=15,(0,1,1); 0.

ŠȠŠ۠

 ՠ

Def: V ,,V=(,)R 1.(+,z)=(,)+(,z);

2.(,+z)=(,)+(,z), ,VR (,).

A = ij i, j =1 (,) ࠠ,蠠,V(,)=(,), ,,V(,)=(,).

(ij=aji), 頖(ij=aji).

. (,)=, (,). . , .,,,, . ,, .

.

), ;

)(),蠠 (,)>0 ((,)0蠠(,)0i=1,2,,n;

),(,), 10, 30, 2 =3>0, 3 =detA =3 (a1). , , 3 >0,..蠠>1.

8. , , , .

. (, ) , .

1.(,)=(,).2.(+,z)=(x,z)+(,z).3.(,y)=(x,y). 4.(,x)>0,蠠x ,(,x)=0,蠠x=. 2 3 , (, ) .1 3, , .,1 2 :(, + z)=( + z, x)=(, )+ (z, x)=(, )+(, z), 13,: , 1 3 , (, ).4 (,):(,) 0,(,)=0=.

. , , , . 4. A( P,Q ) = ak b j.,A(P,Q) :1=1,2=,3=2,4=3 . 2. A(P,Q),PQ .1,2,3 A(P,Q)PQ:

a0,a1,a2,b0,b1,b2РQ1,2,3;) 2 3 0 ; A(P,Q)= 2a 0b0 + a1b1 + a 2b2,堠a0,a1,a2,b0,b1,b2РQ 堠1,2,3. 3. n:

)11(n=1);)11(n=2);

)2111221522(n=2);

)12313+723+21331+732+33(n=2). 2x1x 2 6x1x3 + 14x2x3 + x3. 4.蠠(,):

) A( x, y ) = x1y1 + x 2 y 2 + x3 y 3 2x1y 2 2x 2 y1 + x 2 y 3 + x3 y 2 ;

) A( x, y ) = 3x1y1 + x 2 y 2 x3 y 3 + 9x1y 2 + 9x 2 y1 + 10 1y 3 + 10 3 y1 ; ) A( x, y ) = x1y1 x 2 y 2 + x1y 2 + x 2 y1 3x1y 3 3x3 y1 + 2x 2 y 3 + 2x3 y 2. 5.n :

) 3x1 (n=1);)1812+ 9x2 (n=2);

17. )311;)9(1221+22);)11+212+221+213+231+522+623+632+733; )211312321322. 6.:

) 5x1 4x1x 2 + 8x2 ;) 4x1 + 10x1x 2 4x 2 ;) 4x1 + 4x1x 2 + 8x1x3 3x 2 + 4x3 ; ) 2 x1 x1x 2 + x 2 x 2 x3 x 2x 4 + x3 + x 4. 7.., .:

) 1 + 2 ;2,2,0,+;) 1 2 ;2,1,1,?;) 1 2 ;2,1,1.?;) 1 ;1,1,0,+=;

) 1 2 ;2,0,2,;) 1 ;1,0,1,=;) 1 + 2 3 ;3,2,1,?;) 1 + 2 + 3 ; 3,3,0,+;) 1 2 3 ;3,1,2,?;) 1 2 3 ;3,1,2,?;) 1 + 2 ;2,2,0,+=;

6,4,2,?;) 1 ;1,1,0,+=;) 1 2 3 ;3,0,3,. 7. ,ࠠ(,),ࠠ ,=,Ѡ .

9. A (1,2,,n)蠠(1,2,,n) Vn1,2,,n ,n. ?

(aik=aki) A( x, x ) = i aik i k (ei,ek)=aiki,k=1,2,,n. 10. :

)1+,1;)3+;)3+,1;)2+,2;)1+,1. 11.:

)12+13+14+23+24+34;

) y1 + y2 y3 ;) y1 y 2 y3 ;) y1 y 2 ;) y1 y 2 y3 y 4 ;) y1 + y 2 y3. 12.,, .(,, ):

) 12 1 3x2 12 3 + 12 1x2 24 1x3 + 8x2x3 ;)12+23+34+41;

) y1 + y 2 y3,1=1 2+ 3,2= 2 3,3= 3;) y1 + y 2 y 3,1= 1+2, 2=2+3,3=2+3;) y1 y 2 y 3,1=123,2=1+23,3=3; ) y1 + y 2 y 3,1= 1 2+ 3,2= 2+ 3,3= 2+ 3; ) y1 + y2 y3,1= 1 2+ 3,2= 1+ 2,3= 1+ 2;) y1 y 2, 1=123,2=1+23,3=3,x4=4. 13.:

););););). 14., :

)>2;) 1, 0蠠=1,0).

10. ( )().

Def. m ( m )  ,Bm=A.

11. ( 0), m B = m A.   m k, k .

Def. L(V, V) , V (, ) = (, ), : )堖;) 堖.

L(V, V) Ҡ.

(),,*=1(*=).

Def.L(V,V),*=*.

: .

, * V Ҡ.

, *.

, Ҡ.

, , L(V,V), .

Ҡ.

.

Def. , aijC, , *=*=.,aijR,, ==.

Ҡ.

, .   ,,,, .

.( ).ʠ,. Def. Gk() k : 0 1 L Def. : Ҡ.

{ ek } k =1, . :

3) . , ;

4) k k, ;

5) , ,: ) , . . N(A) )();

)()N();

) xi () N()1 yi1 ,: yi1 = xi,2 yi 2 ,: yi 2 = yi1,..,.

:

) xi () N() :1,y11,y12,,2,y21,y22,,,yp1,yp2.

) p+1, p+2,, l, ()N(),. *:, ,, k .:

1,y11,y12,,2,y21,y22,,,yp1,yp2.,p+1,p+2,,l. (k).

V2 ( ). ?

, 1+ i , ?

, ?

, {ei} in=1 {fi} i=1 . ,?

:2,0,1,1.

, ,??

*, ?

*,?

, , ?

, 10.

, ?

蠠ࠠx(Ax, 11.

x)=1+i,?

1 + i 12.

?

1. {ei} ,*,ࠠà {ei}. * .*? . { ei } i =1 , x = xi ei, y = y j e j , a ki , a kj *, ,:

Aei = *:

xi Aei, yj ej = xi ei, yj A ej xi akiek, yj ej = xi ei, yj a kjek mn = ( em, en ) .

:

a *. { ei } i =1 , A* = A T.

) A* = 1AT , ) * = A . 2. . :

. , 1:

A* = 1AT. , ,, :

1 = i 2 2i .:

*, .

. 3. à.*:

. , 1:

A* = 1AT.  : A* = 1AT. , , , : = 9 5 2 .:

3 5 1. 4.{e1,e2}n. ,{f1, f2} n,: A 1 i 1 i.*{fi},f1=1+2,f2= 1i2. .f1=1+2, f2=1i2: 頠 A* = 1 AT. . 5. . (, ). *, : A = 2 3 1 ;(,)=2111221+2222332+33. . (, ) , , ij 蠠ij,頠.

= 1 2 5 1 2 1 = 0 8 7.

6. 頠 ࠠ⠠ {f1,f2,f3};{e1,e2,e3}. ,: f1= 1, f2=1+ 2, f3= 1 2 +3 ; .f1,f2,f3{e1,e2,e3} ,{e1, e2, e3} {f1,f2,f3}.

P = 0 1 1 .

. , ,,Ҡ=.

( ) Ae = . .

. 7. 1, 2 b1, b2, . , 1(3,4), 2(1, 3), b1(5,0),b2(3,1)?

.1(3,4),2(1,3), b1(5,0), b2(3,1) , A B , AA = B A = BA1.

= ,.. . . . . 8. , : 1= (1ie2),Ae2= (2ie1+22i3),Ae3= (e12i2+23).

. , ⠠1,2,3:

A = * , ..

(): AA* = E.

. . . 9. 1 2. , M1 1 1 0, .蠠1=2.. . () : AA* = E .. . 10. , : 1 2 2.

. :

(,.. = 1), = 1. ( A E ) X = 0 . .

1( dim = n rangA 3 2 = 1). ,,(1,1,0)., , e= (1, 1, 0), = 1. ) 1= 11. .

젠:det(AE)=0. :

: ( + 2 + 1)( 2 + 1) =0,,1,2=1;3,4=1. = 1, . . 1. 1. , x3 = x4 = 0.ࠠ, x1 = x2 = 1. f1 (1, 1, 0, 0) , = 1 1.1: ,⠠ f2, f3, f4 : f3+f4f2=(0,0,0,1),1: {1(1,1,0,0),2(0,1,1,1),3(0,0,0,1).

, f1 1 ,, ࠠ1. = 1 , , , . , g1 1 (1,1, 0, 0), , 1 (1,1, 0, 0). 頠 A1 y = 1 g2(1,3/4,0,0) , = 1 . , , 1 (1,1, 0, 0) , A1y = g2 ., = 12 :

,:

Ag 1 = g 1, Ag 2 = g1 g 2.

) =1,, 1=+: ,.. 蠠=1. f1 (1, 1, 1, 1) 1 , =1.

ࠠ(1)ࠠ1. ,f1=f2+f3+f4, : A1f1.

, 2,. g3 1(1,1,1,1), 1(1,1,1,1). A1g4 = 1 g4(0, 1/2, 0, 1/2) , =1.1(1,1,1, 1) , A1y = g4 . , = 1 2 ,:

,:Ag3=g1,Ag4=g3+g4.ࠠ 4.蠠 1. .*.

* 蠠. 2. , f(x)*, *=. 3. 2 , (p, q) = 00+ 11+ 22, p(t) = 0 + 1t + 2t2, q(t) = 0 + 1t + 2t2, Ap ( t ) = p ( t ).*:

)1,t,t2;)1,t,3t21.

4. ( p, q ) = p ( x ) q ( x ) dx Ap ( x ) = p ( x ).*:

)1,x,x2;)1,x,3x21.

5.  n ( p,q) = p( x ) q( x ) dx Ap( x ) = p( x ). * *=; a2k, 2k+1 = k, a2k+1, k = k (k=1,2,n), 0. 6. *, A = x , f(1) = 0, :

7. à.*:

) 8.{e1,e2}, f1 = e1, f2 = e1 + e2 1 1. * {f1,f2}.

9. {e1, e2, , en} n. , {f1, f2,,fn} n. *{fi}, :

)f1=1,f2=1+2; A 1 1 ;)f1=1,f2=21+2; A 1 3 ;

)f1=1+2+3,f2=2+3,f3=23; A 1 0 0.

10. {e1, e2,, en} n. , {f1, f2,,fn} n. *{fi}, :

)f1=123,f2=1+2+3,f3=3; A 1 1 1 ;

)f1=1+2,f2=2+3,f3=1+3; )f1=1,f2=i1+2,f3=i1+i2+3; A 0 0 1.

11. f1(1,2,1),f2(1,1,2),f3(1,1,0) 0 5 1. * , f1, f2, f3 . 30 30 14. 12. 2 1, 2 b1, b2 ., i bi,. *:

)1(0,1),2(1,3),b1(3,1),b2(2,3);)1(1,1),2(1,4),b1(0,2),b2(3,7).

) 3 1 ;) 1 3. 13. *, {1,2,3},1(0,0,1), 2(0, 1, 1), 3(1, 1, 1) b1(1, 2, 1), b2(3, 1, 2), b3(7, 1, 4) .{1,2,3}. 14.. (,).*, :

0 1 ;(,)=11+522+633+213+231+323+332; 15. (, ) (). *:

)(,)=11212221+522; A 1 3 ; )(,)=11+12+21+322; A 1 1 ; 16. (, ) (). *:

17. , . ࠠ:

);). .



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