陕西科技大学：线性代数（吴明鑫）：di4zhang

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·? TT TT12(1,1,0) (0,1,1) (1 0,1 1,0 1) (1,1)?=? ==?vv 0,
T TT TT
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2.A? 1233()2()5()?++=+vv vv vvè" v,J
T1 (2,5,1,3)=v è T2 (10,1 5,10)=v è è
T
3 (4,1,1,1)=v
·?+ 1233()2()5()?++=+vv vv vvH)úk
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(A) 12,,...,mvvv
M2Lê ?G￡×
(B) 12,,...,mvvv,X?)?t ?G￡úG￡?!¨_×
(C) 12,,...,mvvv
?)?t ?G￡á6?>J- ?G￡4??></×
(D) 12,,...,mvvv
Y?F?ú4??′G
?
2) 12,,...,mvvv? ?G￡4?èí ( c )
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(B)íb??á<Lê,X 12,,...,mkkk
Y 11 22,.,mmkkk+++=v v v0èí 12,,...,mvvv4??
′G
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(D)81 120 0,.0m+++=vv v0èí 12,,...,mvvv4??′G
3) 12 12,,...,,,...,mmvvvuuu
` ? ?G￡4?è,ü?4?á<Lê,XD 12,,...,mkkk
`12,,...,mlll
S 1 11 2 22( ) ( ),.,( )m mmkl kl kl++++++=v v v0èí
(A) 12 12,,...,,,...,mmvvvuuu
` F?4??′G
(B) 12 12,,...,,,...,vvvuuu
` F?4??,ìG
(C) 1 12 2 1 12 2,,...,,,...,mm mm++vuvuvuvuvuvu+,4??′G
(D) 1 12 2 1 12 2,,...,,,...,mm mm++vuvuvuvuvuvu+,4??,ìG
4) ?G￡4? 123,,vvv4??′Gèí1? ?G￡4?4??′G,X ( )
(A) 122331,,+++vvvvvv (B)12231 23,,2+ + ++vvvvv vv
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??ít ?G￡F?4??′G
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,ü?t ?G￡èá6?*üJ- ?G￡4??></
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(B) 12233441,,,vvvvvvvv4??′G
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?G￡4? 12,,...,muuu34??′G
A?a?A? 11 22 rmkkk+++=u u uoLí
1122()()()m m pmpmmkkkk kkk++ + ++ ++ ++ ++ =v v v voL LLLL
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11221
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9.A? ?G￡4? A? 12,,...,mvvv,X/? 1r è ?G￡4? B? 12,,...,tuuu,X/? 2r è ?G￡4? C?
12 12,,...,,,,...,mtvvvuuu,X/? 3r èA?a 12 312max{,}rr rrr≤≤+
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2 12 12 12{ (,,...,) |,,...,+ +...+ 1}V xx x xx x x x xè èK? V 1
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1V  ?G￡0NKèè′?
T
12 12(,)0nnxxxxx= +++=x LL
T
12 12yyyyy= +++=y
TT
12 12 1 12 2(,,,) (,,,) (,,,)n n nnxx x yy y xyxy xy+= + =+++xy LLL
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12 12( )( )0nnxxxyyy= +++ + + ++ =LLè 1+∈xyV
TT
12 12,(,,,) (,,,)xxx xxxλ λλλλ∈=Rxxè=
1 2 12( )0nnxx xxxxλλ λλ+ ++ = +++ =LLè 1λ ∈xV
2V á ?G￡0NKèè′?
T
12 12(,)1nnxxxxx= +++=x LL
T
12 12yyyyy= +++=y
11 22()()()nnxy xy xy++++++L
12 12( )( )112nnxxxyyy= +++ + + ++ =+=LLè 2+?xyV
TT
12 12,(,,,) (,,,)xxx xxxλ λλλλ∈=Rxxè=
1 2 12( )1nnxx xxxxλλ λλ λλ+ + + = + + + =?=LLè 2λ?xV
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A?a?A? 123(,,)=A vvv
1
123
011 110
,,101(1)101 20
110 011
= = =? =?≠A vvv
b R(A)=3 123,,vvv4??′G
?+b 123,,vvv
Y4èèè/? 3è1 123,,vvv!8
Y4è0NKè,X?4?
è+ 123,,vvv*ó?,X ?G￡0NKè 3R
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A?a?A? { } { }1 11 22 1 2 2 13 24 1 2,,kkRRλ λ λλ==+∈==+∈Vxvv Vyvv
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112011201120
1011 0131 0131 =
013 1 01 3 1 0000
013 1 01 3 1 0000




vvvv,:
1110
10 11 22
013 1 3101
220000
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0000






::
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(1)
1 2 34
1 2 34
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1234
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35420
87630
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a340,4xx==k120,1xx==
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40
01,
10
34

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(2)
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01
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3
4
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x xx
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TT
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?
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0 n
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A??0?ê
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?
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1 2 34
1 2 34
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5361
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x x xx
x x xx
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+ + + =
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=

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11005 10108
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8 1 18
13 1 1 13,
0 1 10
2 0 02
kk

∴= = +=+



 [E·? =
(2)
1 5 2 3 11 1 5 2 3 11
5 3 6 1 1 ~ 0 28 4 14 56
2421 6 014 27 28

=

B
91101
1 5 2 3 11 72
11 11~01 2~01 2
72 72
00000 00000






1 2 12
1 9 1 9 11
2 1 1 1 12,,
0 7 0 7 00
0 0 2 0 20
kk

∴= = =?=++



x
23êA? η*M2U$?4???/?4? A x = b,X?t?·è 12,,...,nr? íh,XU$?4???/?4?,X
?t
.?·2?èA?a?
(1) * 12,,,...,nr? 4??′G×
(2) * **12,,,...,nr?+++4??′G
A?a (1) ?A?"?A? * 12,,,...,nr? 4??,ìGí,ü-á< ,XD 01,,...,nrccc?S
k???0?,
*
0 11 22,.,nr nrcccc++++= 0 (1)
J
,0 0c ≠ úí,12,,...,nr? 4??,ìG5àa
.?·2?á4??,ìG,X{*ó-ˉ,ò
?
+b * (M?·è 12,,...,nr? 
.?·2?èk
* **
012 000
0*
0 11 22
(,.,)
(,.,)
nr nr
nr nr
ccc c cccc
cccc


++++ ====
++++=?
$   $ $ b b0
$   0
5à00c≠è =b0è5àNl
A1?/?4?M2U$?4???/?4?k ≠b0
{*ó-ˉ,òA?á?0?, * 12,,,...,nr? 4??′G,
(2) ?A?"?S * **12,,,...,nr?+++4??,ìG
?
í,ü-á<Lê,XD 01,,...,nrccc?Sk???0?,
****
0 11 22( ) ( ),.,( )nr nrcccc+ ++ +++ += 0 ?2?
G0 1 11()nr nr nrccccc+++ + ++ = 0LL
1) 81 01 0nrccc?+++=L +b 12,,...,nr? 4??′G,X?4?
.?·2?
1 100 0 0nr nrcccccc== = +++ =?=LL ?!8ê 01 0nrccc?====L
aA?-ˉ,ò,
2) 81 01 0nrccc?+++≠L è+Nl (1)-1 * 12,,,...,nr? 4??′G
0112000nr nrcc c cc c c+++ ==== =?=LL aA?-ˉ,ò
?
4DTE?A?á?0? s
QNlkA?,
24êA? 12,,...,s M2U$?4???/?4? A x = b,X st?·è
12 12,,...,,+ +...+ =1sskkkRkkk∈è èA?a 11 22,.,sskkk= + ++x    3W,X?·
?
A?a?+b 12,,...,s M2U$?4???/?4? A x = b,X st?·,
Y ( 1,,)i is==$b L
5à121221(,.,) ( )s ssskk kkk k kk= + ++ = ++ =$++ $ $ $ bbLL
G 11 22()sskkk+++=$ bL
1 11 22 sskkk+ ++  L 3?/?,X?·ê
25êA?M2U$?4???/?4? A x = b,X2?D-?L ,X/? rè 121,,...,nr?+  W,X n?r+1t4?
?′G,X?·èA?A?aWY?)?t?·
 ?></
12211121..,(,.,1)nr nr nrkk k kkk?+?+?+=+++ +++=x   
A?a A? x =Axb,X?·ê
+NlA?-1? 121,,...,nr?+  4??′Gè
 =Axb,X?·ê
a 1 212 313 41 11,,,...,nr nr+=? =? =? =? èíWà
 =Ax0,X
·êè 12,,...,nr? 4??′GêV81á'
èí,üá<Lê,XD?
12,,...,nrlll?Sk 11 22,.,nr nrlll+++= 0
G121 231 11( ) ( ),.,( )nr nrlll+?+?++?=  0
z G 121231(,.,),..nr nr nrllllll+?+++++++=    0
+ 121,,...,nr?+  4??′G-1 12 12( ),.,0nr nrllllll+++ =====L
1 12,,...,nr? 4??′Gè =Ax0,X?4?
ê
+b 1,x 
 =Axb,X?·è1 1?x  ,X =Ax0?· 1x  ?+ 12,,...,nr? 4??
><ê G
121321..,nr nrkkk = + ++[  
221 331 1 11()().()nr nrkkk?+?+=?+?++?  
1 23 12233 11(1,.,),..nr nr nrkkkkkk?+?+?+= + + ++[   
?123112311,..,.,1nr nrkkkk kkkk?+?+=++++=
G? 112233 11..,nr nrkkkk?+?+= + + ++x
?A?!?
26ê?-1 ?G￡4?è A? TTT123=(0 1 2 3) =(3 0 1 2) =(2 3 0 1)aaaèè×
B?T TT=(2 1 1 2) =(0 2 1 1 ) =(4 4 1 3)?bbbèè
A?a B4?6?+ A4?4??></è A4?á6?+ B4?4??></
A?a?+b 123123
0322 0 4 10 3 1 2 4
1031 24 03 2 2 0 4(,,)~
2101 1 1 01 6 1 5 7
3212 1 3 02 8 17 9

=


aaabbb
1 123 123
111
444
111(,,)(,,)
222
1 35
4 44


=


bbb aaa è ?-?L
111
444
111
222
1 35
4 44





á ?Eúè1 ?G￡4?
Aá6?> ?G￡4? B></
?
27ê?-1 ?G￡4? A? TT12=(0 1 1) =(1 1 0)aaè×
B?TTT1 23=( 1 0 1) =(1 2 1) =(3 2 1)b bbèè
A?a ?G￡4? Aa ?G￡4? B1?
?
A?a?+b 12123
01 113 00000
(,,,,) 11022~01 113
1011 1 1011 1

=?

aabbb
11102
2210111
11~01 113~ 011
2200000
0 0000





1? 123 12 111(,,)(,)113=bbb aa × 12 123
11
22
11(,)(,,)
22
00


=


aa bbb
28ê?-1 123 234(,,)2 (,,)3RR==aaa aaaè èA?a
(1) 1a 6?+ 23,aa4??></× (2) 4a á6?+ 123,,aaa4??></
A?a? (1)′ 123 234(,,)2 (,,)3RR==aaa aaaè è1 ?G￡4? 123,,aaa4??,ìGè5à
?G￡4? 234,,aaa4??′GèEˉ5à ?G￡4? 23,aa34??′Gè 1a 6?+ 23,aa4??></
(2)A? 4a 6?+ 123,,aaa4??></è ?+b 1a 6?+ 23,aa4??></è1 4a 36?+ 23,aa
4??></èí 234,,aaa4??,ìGèE-a 234,,aaa4??′G-ˉ,òè 4a á6?+ 123,,aaa4?
?></
?