资源描述:
《线性代数期中试卷.doc》由会员上传分享,免费在线阅读,更多相关内容在教育资源-天天文库。
1、Linearalgebraquestions2010/12Papercode:10043CClassTimes:45ClassesCourseTitle:LinearAlgebraClass:ElectiveclassesQ.1Q.2Q.3Q.4Q.5Q.6TotalScores【Note:Writeyouranswersandthequestionnumbersonyouranswersheet!Otherwise,invalid!】Q.1/Scores1.Fillintheblank(2*10=20scores)(1)LetA=(1,–1,0)andB=
2、(1,1,1,1,2),soRank(ATB)=(1).Solution:(2)LetAisa(3×3)matrix,
3、A
4、=−3,so
5、−3A
6、=(81)(3)IfAisan(n×n)matrix,X1andX2arethesolutionstothelinearequationsAX=B,(X1≠X2),so
7、A
8、=(0).(4)LetAbealinearlydependentthevectorsset,A={β1,β2,β3},hereβ1=(1,2,3),β2=(4,t,3),β3=(0,0,1),sot=(8).Solution:(5)Therea
9、rendimensionvectors:α1,α2,α3,if(αi,αj)=i+j,sowehave(α1+α2,α1–α3)=(4).Solution:For(αi,αj)=i+j,sowehave(6)IfX1=(1,0,2)TandX2=(3,4,5)Taretwosolutionstothethreevariablesnon-homogeneouslinearequationsAX=B,sothecorrespondinghomogeneouslinearequationsAX=0hassolution((2,4,3)T).Solution:X1–
10、X2=(1,0,2)T–(3,4,5)T=(2,4,3)TisasolutiontotheequationAX=0.(7)Letβ=(1,–1,0,2)Tandγ=(a,1,–1,1)Tareorthogonalvectors,soa=(–1).Solution:βandγareorthogonalvectors,soβ·γ=0,thatis(8)A,Bare(4×4)matrices,forA=(α,α1,α2,α3)andB=(β,α1,α2,α3),if
11、A
12、=1and
13、B
14、=2,sowecanfindthat
15、A+B
16、=(24).Solution:A
17、+B=(α,α1,α2,α3)+(β,α1,α2,α3)=(α+β,2α1,2α2,2α3),so
18、A+B
19、=det(α+β,2α1,2α2,2α3)=det(α,2α1,2α2,2α3)+det(β,2α1,2α2,2α3)=23det(α,α1,α2,α3)+23det(β,α1,α2,α3)=8+16=24(9)LetAis(3×3)matrices,for
20、A
21、=2,so
22、A*-3A-1
23、=(–1/2).Solution:for
24、A
25、=2,
26、A-1
27、=1/2A*=A-1
28、A
29、
30、A*-3A-1
31、=
32、A-1
33、A
34、-3A-1
35、=
36、A-1(
37、A
38、-3)
39、=
40、
41、(-1)A-1
42、=–1/2(10)Let,
43、A
44、≠0,soA−1=().Solution:A-1=A*/
45、A
46、Q.2/Scores2.MultipleChoice(1*20=20scores)(1)LetA,B,C,andDare(n×n)matrices,Iisan(n×n)identitymatrix,thefollowingstatementiscorrect(D).A.IfA2=0,thatisA=0B.IfA2=A,sowehaveA=0orA=IC.IfAB=AC,andA≠0,soB=CD.IfAB=BA,wehave(A+B)2=A2+2AB
47、+B2(2)Ifthevectorsgroupβ1,β2,andβ3arelinearlyindependent,andβ1,β2,andβ4arelinearlydependent,so(C).(A)β1canbeexpressedasalinearcombinationbythevectorsβ2,β3andβ4.(B)β2can’tbeexpressedasalinearcombinationbythevectorsβ1,β3andβ4.(C)β4canbeexpressedasalinearcombinationbythevectorsβ1,β2an
48、dβ3.(D)β4can’tbeexpresseda