|
|
??????????????????
??? ?????
4-1 ?????
?????
1?????????? C?
A.n??????????n??? B.n??????????n?1???
C.n???????????????????
D.n?????????????????????. 2?????????? B?
A.?D?0???????????????? B.?D?0???????????????? C.????????????????????D?0? D.??????????????D?0. ?????
??x1?x2?x3?0?
1???????????x1??x2?x3?0?????
?x?2?x?x?0
23?1
??? 1??? 0.
2??????????????????????????D?0, ????????xi?DiD
.
???????????????
?8x?3y?2
1??
6x?2y?3?
??
D?
86
32
??2?0
??5
D2?
86
23??12
D1?
23
32
?
D1D?52,y?
???x?
D2D
??6
?x1?2x2?x3??2?
2??2x1?x2?3x3?1
??x?x?x?0
23?1
1D?2
?2111
1?3?1
r2?2r1r3?r1
100011010011
?23?1?5
1
?5??5?00
??
?2D1?1
01D2?2
?11D3?2
?1
?1?211?210?211D1D
?3r1?2r21?11
05
?3??5?1?5?3??10?101??50D3D
?
?3r1?2r22?1?2
?15
?
1r1?2r220?1,x2?
?1D2D
?1
??? x1??2,x3?
?2x?z?1?
3??2x?4y?z?1
??x?8y?3z?2?
2D?2
048?1
?10048
?1
?1?20?03
??
1D1?1
22D2?2
?12D3?2
?1
?1c1?2c303
1
0482
50
?1048112048D1D
?1c3?c113?1
2
0?205112048
00?0511?202D3D
?
?1c3?c2231
?10
?
1c1?2c302?1,y?
?5D2D
?1
??? x??0,z?
?x1?x2?x3?x4?5?
?x1?2x2?x3?4x4??24??
?2x1?3x2?x3?5x4??2?3x?x?2x?11x?0
234?1
??
1D?
1231??5?2
12?31?2?3?15D1?
?2?205??2?2
1D2?
123?7??12?151D3?
123?2??511
5?2?212?31?1?555?2?20?2?3?15?2?20?10?1828
1?1?123?7812?31
14?511
r2?r1r3?2r1r4?3r1100
1000
11?5?23
8??142145
12?310
32??142?225?7?12?1500?112?310?2951
07?27
??426?13?31??28485?2?20
?10?182801?2?3?1
13?78?1?550
?10?18280
1?2?3?1
13?78
r2?5r1r3?2r11?1?12
1
?2?13?5
4c3?2c2?2?5c4?11c2?211
0?27231000
0?1?35
?10?18281?1?123?781
c1?5c2c3?10c2
14?511
r2?r1r3?2r1r4?3r1
2333?15?2?5
r1?2r3r2?3r3
4c1?3c2112c2?5c11c3?5c1
2?2?511
?5c4?11c211
1D4?
123?2??511
?1?55
12?315?2?2
D1D
1?1?12
5?212?315
?1?550
5?2?20
?2c1?3c2?5?2c3?2c2110
?256
D2D
0?100
r3?r2r2?5r1
?27?142?4
D3D
?3,x4?
D4D
??1
??? x1??1,x2??2,x3?
4-2 ???????
?????
1???m?n??A???n?1??1,?2????????AX?0 ????????k ??????????AX?0????? D ?. A.k?1? B.k?2?C.k(?1??2)? D.k(?1??2).
????m?n??A???n?1??????AX?0????? ?1??????1,?2????????AX?0???????? ???1??2?0?AX?0???????AX?0????k(?1??2)?
?kx1?x2?x3?0?
2????????x1?kx2?x3?0 ??????????? C ?
?2x?x?x?0
23?1
A.k???0? B. k???1?
C. k?0?1? D.???k????.
?a1b1?a2b1
3?A??
?M??anb1
a1b2a2b2Manb2
LLOL
a1bn?
?a2bn
?,?a?0(i?1,2,L,n)??0(j?1,2,L,n)? bij?M?anbn?
?Ax?0?????????A????.
A.n?1? B.n? C.1? D.???.
?a1b1?a2b1
????A??
?M??anb1
a1b2a2b2Manb2
LLOL
a1bn??a1?
???a2bna
???2??b
1
M??M????anbn??an?
b2
L
bn?
???R(A)?1??a1b1?0?R(A)?1????R(A)?1?
4??A?n????r(A)?n?3 ??a1,a2,a3?Ax?0??? ??????????Ax?0??????? A??
A?a1?a2,a2?a3,a3?a1?B?a2?a1,a3?a2,a1?a3? C?2a2?a1,?????
1?n????????Am?nX?0????????????R(A)?n .
?(1??)x1?2x2?4x3?0
?
2????0???2???3??????????2x1?(3??)x2?x3?0????.
?x?x?(1??)x?0
23?1
12
a3?a2,a1?a3? D?a1?a2?a3,a3?a2,?a1?2a3?
3???????????1???2,1,0???2??3,
T
0,
1????
T
???????___ __x1?2x2?3x3?0.
?x1??2x2?3x3
?
????????x2?x2
?x?x3?3
?x1?2x2?3x3?0
??????????????????(?)
??? ?????
4-1 ?????
?????
1?????????? C?
A.n??????????n??? B.n??????????n?1???
C.n???????????????????
D.n?????????????????????. 2?????????? B?
A.?D?0???????????????? B.?D?0???????????????? C.????????????????????D?0? D.??????????????D?0. ?????
??x1?x2?x3?0?
1???????????x1??x2?x3?0?????
?x?2?x?x?0
23?1
??? 1??? 0.
2??????????????????????????D?0, ????????xi?DiD
.
???????????????
?8x?3y?2
1??
6x?2y?3?
??
D?
86
32
??2?0
??5
D2?
86
23??12
D1?
23
32
?
D1D?52,y?
???x?
D2D
??6
?x1?2x2?x3??2?
2??2x1?x2?3x3?1
??x?x?x?0
23?1
1D?2
?2111
1?3?1
r2?2r1r3?r1
100011010011
?23?1?5
1
?5??5?00
??
?2D1?1
01D2?2
?11D3?2
?1
?1?211?210?211D1D
?3r1?2r21?11
05
?3??5?1?5?3??10?101??50D3D
?
?3r1?2r22?1?2
?15
?
1r1?2r220?1,x2?
?1D2D
?1
??? x1??2,x3?
?2x?z?1?
3??2x?4y?z?1
??x?8y?3z?2?
2D?2
048?1
?10048
?1
?1?20?03
??
1D1?1
22D2?2
?12D3?2
?1
?1c1?2c303
1
0482
50
?1048112048D1D
?1c3?c113?1
2
0?205112048
00?0511?202D3D
?
?1c3?c2231
?10
?
1c1?2c302?1,y?
?5D2D
?1
??? x??0,z?
?x1?x2?x3?x4?5?
?x1?2x2?x3?4x4??24??
?2x1?3x2?x3?5x4??2?3x?x?2x?11x?0
234?1
??
1D?
1231??5?2
12?31?2?3?15D1?
?2?205??2?2
1D2?
123?7??12?151D3?
123?2??511
5?2?212?31?1?555?2?20?2?3?15?2?20?10?1828
1?1?123?7812?31
14?511
r2?r1r3?2r1r4?3r1100
1000
11?5?23
8??142145
12?310
32??142?225?7?12?1500?112?310?2951
07?27
??426?13?31??28485?2?20
?10?182801?2?3?1
13?78?1?550
?10?18280
1?2?3?1
13?78
r2?5r1r3?2r11?1?12
1
?2?13?5
4c3?2c2?2?5c4?11c2?211
0?27231000
0?1?35
?10?18281?1?123?781
c1?5c2c3?10c2
14?511
r2?r1r3?2r1r4?3r1
2333?15?2?5
r1?2r3r2?3r3
4c1?3c2112c2?5c11c3?5c1
2?2?511
?5c4?11c211
1D4?
123?2??511
?1?55
12?315?2?2
D1D
1?1?12
5?212?315
?1?550
5?2?20
?2c1?3c2?5?2c3?2c2110
?256
D2D
0?100
r3?r2r2?5r1
?27?142?4
D3D
?3,x4?
D4D
??1
??? x1??1,x2??2,x3?
4-2 ???????
?????
1???m?n??A???n?1??1,?2????????AX?0 ????????k ??????????AX?0????? D ?. A.k?1? B.k?2?C.k(?1??2)? D.k(?1??2).
????m?n??A???n?1??????AX?0????? ?1??????1,?2????????AX?0???????? ???1??2?0?AX?0???????AX?0????k(?1??2)?
?kx1?x2?x3?0?
2????????x1?kx2?x3?0 ??????????? C ?
?2x?x?x?0
23?1
A.k???0? B. k???1?
C. k?0?1? D.???k????.
?a1b1?a2b1
3?A??
?M??anb1
a1b2a2b2Manb2
LLOL
a1bn?
?a2bn
?,?a?0(i?1,2,L,n)??0(j?1,2,L,n)? bij?M?anbn?
?Ax?0?????????A????.
A.n?1? B.n? C.1? D.???.
?a1b1?a2b1
????A??
?M??anb1
a1b2a2b2Manb2
LLOL
a1bn??a1?
???a2bna
???2??b
1
M??M????anbn??an?
b2
L
bn?
???R(A)?1??a1b1?0?R(A)?1????R(A)?1?
4??A?n????r(A)?n?3 ??a1,a2,a3?Ax?0??? ??????????Ax?0??????? A??
A?a1?a2,a2?a3,a3?a1?B?a2?a1,a3?a2,a1?a3? C?2a2?a1,?????
1?n????????Am?nX?0????????????R(A)?n .
?(1??)x1?2x2?4x3?0
?
2????0???2???3??????????2x1?(3??)x2?x3?0????.
?x?x?(1??)x?0
23?1
12
a3?a2,a1?a3? D?a1?a2?a3,a3?a2,?a1?2a3?
3???????????1???2,1,0???2??3,
T
0,
1????
T
???????___ __x1?2x2?3x3?0.
?x1??2x2?3x3
?
????????x2?x2
?x?x3?3
?x1?2x2?3x3?0
???????????????
???
? ? ?
??4?1
1???????????
2
?1?f(x1,x2,x3)=2x12?x2?4x1x3?2x2x3?
?2?f(x1,x2,x3,x4)=2x1x2?2x1x3?2x1x4?2x3x4?
???1???
?2
?
f(x1,x2,x3)=(x1,x2,x3)?0
?2??2?
?????f(x1,x2,x3)??????0
?2?
0?1?1
0?1?1
2???1?0??
?x1
??x2?x?3
???, ??
2???1?? 0??
?2???
?0??1
f(x1,x2,x3,x4)=(x1,x2,x3,x4)?
1??1??0??1
?????f(x1,x2,x3,x4)??????
1??1?
1000
1001
1000
1001
1??0?1??0??
?x1
??x2?x?3?x?4????? ???
1??0?? ?1?0??
2????????????????? ?
?1?
1
?1???
?2?1??2
12
??0
1?
??
2??1
?
?2?? ?2?2
??
??1?
2?????0
?
12?11212
?112012
?0??1?2?? 1??2?1???
?
0?2
T
???1??X?(x1,x2,x3)??
??1?
1
f(x1,x2,x3)=XTAX=(x1,x2,x3)??
?2?1??2
?
12
0?2
1??2??2?
??2??
?x1??x2?x?3??? ??
=x12?2x32?x1x2?x1x3?4x2x3? ?2??X?(x1,x2,x3,x4)T??
??0??1?
f(x1,x2,x3,x4)=XTAX=(x1,x2,x3,x4)2
???1???0?
12?11212
?112012
?0??1?2?1??2?1???
?x1??x2?x?3?x?4???? ???
22
=?x2?x4?x1x2?2x1x3?x2x3?x2x4?x3x4?
??4?2
1??????????????????????????????
22
?1?f(x1,x2,x3)=2x1?x2?4x1x2?4x2x3?
?2?f(x1,x2,x3)=2x1x2?2x2x3?
222
?3?f(x1,x2,x3)=x1?2x2?3x3?4x1x2?4x2x3?
???1????f(x1,x2,x3)??? ?2
?
A=??2
?0?
?21?2
0???2?? 0??
A??????
??2
det?(E?A)=
20
20
2=(??2)(?2?5??4)=0?
??12
?
????A?????1??2??2?1??3?4?
???1??2????????(?2E?A)X?0?????????
?1?(1,2,2)T?
???2?1????????(E?A)X?0???????????2?(2,1,?2)T?
???3?4????????(4E?A)X?0???????????3?(2,?2,1)T?
??1,?2,?3????? ?1?
1
11
?1?(,
122T
,)? 33321
23
T
?2?
21
?2?(,,?
3323
)?
?3??
3
?3?(,?
21T
,)? 33
?1??32
P=(?1,?2,?3)=?
?3?2??3
23132?3
2??3?2
??? 3?1??3?
??2?
? PTAP=diag(-2,1,4)=?0
?0?
010
0??0?? 4??
?????X=PY??
122?
x?y?y?y3
12?1
333
?212?
?x2?y1?y2?y3?
333?
?x?2y?2y?1y3123?333?
???f(x1,x2,x3)???????
222
?2y1?y2?4y3?
?2?????1??????
?1??2?
P=?0
??1??2
??121212
?
1??2??0
?1?T
?PAP=?0?2?0?
?1?
?2?
020
??0?? ?2??
2
??????2y2?2
2y3?
?3?????1??????? ?2??3
1
P=??
?3?2???3
132?323
2??3??2
?2?T
? PAP=?03??01???3?
050
0??0?? ?1??
222
??????2y1?5y2?y3?
2?????????????????
222
?1?f(x1,x2,x3)=x1?2x2?5x3?2x1x2?2x1x3?6x2x3?
?2?f(x1,x2,x3)=2x1x2?4x1x3?
?3?f(x1,x2,x3)=?4x1x2?2x1x3?2x2x3? ???1?????x1?????
f(x1,x2,x3)=x1+2x1(x2?x3)+(x2?x3)?(x2?x3)+2x2+6x2x3+5x3
222
=(x1?x2?x3)+x2+4x2x3+4x3?
22222
????????x2???????
22222
f(x1,x2,x3)=(x1?x2?x3)+x2+4x2x3+4x3=(x1?x2?x3)+(x2?2x3)?
?1?TT
?Y=(y1,y2,y3)?X=(x1,x2,x3)?B=?0
?0?
2
2
110
1??2?? 0??
?Y=BX?????????????y1?y2?
?2????????????????????????????????????1?
????????
?
?x1?y1?y2?x1??1
????
?x2?y1?y2???x2?=?1
?x?y?x??0
33??3??
1?10
0?
?0?1??
?y1?
??y?2?? ?y??3?
???????
f(x1,x2,x3)=2(y1?y2)(y1?y2)+4(y1?y2)y3
2
=2y12?2y2+4y1y3+4y2y3
=2(y1?y3)2?2(y2?y3)2? ?1
?TT
?Y=(y1,y2,y3)?Z=(z1,z2,z3)?B=?0
?0?
010
1???1?? 0??
2
?Z=BY?????????????2z12?2z2?
?3?????2?????????????????
?4z1?4z2?z3?
2
2
2
3???????????????????
222
?1?f(x1,x2,x3)=x1?2x2?4x3?2x1x2?4x2x3?
222
?2?f(x1,x2,x3)=x1?3x2?x3?2x1x2?2x1x3?6x2x3?
?3?f(x1,x2,x3)=4x1x2?2x1x3?6x2x3???????????????? ???1????f(x1,x2,x3)???? ?1
?
A=?1
?0?
122
0??2?? 4??
112010
0?
?2?4?????0??0?1??
?1
??0?0??1?0??0?
012?110
0??2?4?????0??0?1??
?1
??0?0??1?0??0?
010?110
0??0?0??? 2???2
?1??
??
?1
??1?0?A?
???E??=???1
?0??0?
122010
0??2?4?????0??0?1??
?1
??0?0??1?0??0?
?
????????????????????????
?????http://www.850500.com/news/17436.html
?????,??! |
|