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1/2StudentNumber:YourName:2016—2017FallSemesterUNIVERSITYOFSCIENCE＆TECHNOLOGYBEIJINGLinearAlgebraFinalExamTime:09:00-11:30A.M.FullMark:100YourMark:Notation:Pleasefilloutandsignthefrontofyourexambooklet.Nobooksorelectronicdevicesallowed.Nousinganynotesorformulas!Nocheating!Youmaykeepthispaper.Solutionswillbepostedonthecoursewebsiteaftertheexam.Pleasedonotanswerthefollowingproblemuntilwegivethesignal.1.(20points)LetA=[]Findbasesofthefollowingvectorspacesandstatetheirdimensions.(a)ThecolumnspaceofA.(b)TherowspaceofA.(c)ThenullspaceofA.(d)TheorthogonalcomplementofthecolumnspaceofA.2.(15points)LetA=()(a)ComputeAkforallintegersk0.Writetheanswerasexplicitlyasyoucan,intheformofa-matrixwithentriesdependingonk.(b)Solvetheinitialvalueproblemx'(t)=Ax(t)withx(0)=()3.(10points)(a)Letbethevectorspaceofpolynomialsofdegreelessthanorequalton.LetTbethelineartransformationfromtodefinedbyT()(t)=p(2)+(t–2)̇(t)+t3̈(5t)(YouarenotrequiredtoshowthatTislinear.)FindthematrixofTwithrespecttotheB3={1,t,t2,t3}ofP3andtheB4={1,t,t2,t3,t4}of.(b)Findtheequationy=ax+boftheleast-squareslinethatbestfitsthedatapoints(1,2),(2,2),and(3,4).2/24.(20points)LetA=()(a)FindthesingularvaluesofA.(b)FindtwounitvectorsinR4thatareorthogonaltoeachotherandtothecolumnsofA.(c)FindasingularvaluedecompositionofA.5.(15points)Provethefollowingassertions.Allmatricesinthisproblemarereal-matrices.(a)IfmatricesAandBaresimilar,thentheyhavethesamerank.(b)SupposethematrixAsatisfiesthefollowingconditions:Aissymmetric,A2=A,andrank(A)=1.ThenthereexistsaunitvectoruinRnwiththepropertythatA=uuT.(Hint:whatdoestheconditionA2=AtellyouabouttheeigenvaluesofA?Alsousetheresultofpart(a).)6.(20points)Trueorfalse?Proveyourassertions!Allmatricesinthisproblemarereal.(a)IfAisanmxn-matrix,thenAandATAhavethesamenullspace.(b)Theformula()=∫))))))definesaninnerproductonthespaceofcontinuouslydifferentiablefunctionsontheinterval[0,1].(Afunctioniscalledcontinuouslydifferentiableifexistsandiscontinuous.)(c)IfAisapositivedefinitesymmetric-matrix,thenthereexistsanon-zerovectorxinRnwiththepropertythatxTAx||x||2.(d)IfAisan-matrixandQisanorthogonal-matrix,thenAandAQhavethesamesingularvalues.