MATA22Algebraic Operations

Universityof TorontoScarboroughCampusDepartmentof Computer andMathematicalSciences

ThishomeworkwasreleasedonMon.Jan.8th09:00(EST).Itisdueon Fri.Jan.19th17:00(EST).WeencourageyoutotalktoyourTAsduringtutorial,attendoficehours,andaskprofessorsforhelpwiththisassignment.Youmayusethetextbookwithoutcitingitasareference,howeverallotherbooksandinternetsourcesmustbecited.Pleasesubmityouroriginalwork viaCrowdmark.TheversionofthishomeworkonCrowdmark istheonlyoficialversionoftheassignment.ThisPDF isprovidedforyourreference.PleasechecktheversiononCrowdmarkbeforeuploadingyoursolutions.Ineachhomework assignmentthereisonequestionthatmustbetypedup inlatex.Forthisassignment,thatquestionisQ4.Ineachquestionthathasanunderlinedterm,simplydefiningtheunderlinedtermwillguaranteethatyoureceiveatlease20percentonthegivenquestion.Readings§1.1VectorSpaces§1.2Subspaces

ProblemsQ1.(a)Double additionofrealnumbersisgivenby theformula:a⊞b= 2a+ 2bwhere+ istheusualadditionof realnumbers.Is⊞associative?(b)Halfsyadditionof realnumbersisgivenby theformula:a⊞b= 2a+bwhere+ istheusualadditionof realnumbers.Is⊞associative?Q2.Complicatedmultiplicationofpairsofrealnumbersisgivenby theformula:(a,b)⊡(c,d)= (ac−bd, ad+bc)whereaddition,subtraction,andmultiplicationaretheusualoperationson realnumbers.(a)Determine(withproof)whether⊡isassociative.(b)Determine(withproof)whether⊡iscommutative.Q3.The axiomsofavectorspacerequiretheexistenceofanadditiveinverseforeachelement.For allv∈Vthere isw∈Vsuch thatv+w=0.We say:wistheadditiveinverseofv.Provethefollowing:(a)Additiveinversesareunique.Page2

MAT A22Winter 2024(b)The additiveinverseof theadditiveinverseofvisv.Itmighthelptoreadthequestion as:(The additiveinverseof(theadditiveinverseofv))isv.Q4.(***The solutionforthisquestionmustbe typed usingLatex***)ConsiderR+={x:x >0}.We definethefollowingoperations:x⊞y=xyc⊡x=xcforc∈RShow thatV= (R+,⊞,⊡)isanR-vectorspace.Q5.ConsideranyvectorspaceV.Letx,y,zbeelementsofVanda,bbescalars.(a)Prove:Ifx+y=x+ztheny=z.(b)Prove:(a+b)(x+y)=ax+bx+ay+by.Q6.Consider polynomials with complex coeficients and degree at mostn.Pn(C)={a0+a1x+a2x2+···+anxn:ai∈C}ProvethatPn(C)isavectorspace.(Hint:summationnotationwillsimplifyyourargument.)