2017~2018 Final Exam 线性代数英文试题.doc

StudentNumber:YourName: 2016—2017FallSemesterUNIVERSITYOFSCIENCE&TECHNOLOGYBEIJINGLinearAlgebraFinalExamTime:09:00-11::100YourMark:Notation:!Nocheating!.(20points)LetA=221663-301104013104Findbasesofthefollowingvectorspacesandstatetheirdimensions.(a) ThecolumnspaceofA.(b) TherowspaceofA.(c) ThenullspaceofA.(d) .(15points)LetA=4-130(a) ComputeAkforallintegersk≥,intheformofa2×2-matrixwithentriesdependingonk.(b) Solvetheinitialvalueproblemx'(t)=Ax(t)withx(0)=103.(10points)(a)(p)(t)=p(2)+(t–2)p(t)+t3p(5t)(YouarenotrequiredtoshowthatTislinear.)FindthematrixofTwithrespecttotheB3={1,t,t2,t3}ofP3andtheB4={1,t,t2,t3,t4}(b)Findtheequationy=ax+boftheleast-hatbestfitsthedatapoints(1,2),(2,2),and(3,4).4.(20points)LetA=23101021(a) FindthesingularvaluesofA.(b) FindtwounitvectorsinR4thatareorthogonaltoeachotherandtothecolumnsofA.(c) .(15points)×n-matrices.(a) IfmatricesAandBaresimilar,thentheyhavethesam