线性代数英文ppt1

Chapter1LinearEquationsandVectorsLinearAlgebraDefinition•Anequationinthevariablesxandythatcanbewrittenintheformax+by=c,wherea,b,andcarerealconstants(aandbnotbothzero),iscalledalinearequation.•Thegraphofthisequationisastraightlineinthex-yplane.•Apairofvaluesofxandythatsatisfytheequationiscalledasolution.1.1MatricesandSystemsofLinearEquationssystemoflinearequationsCh1_2Figure1.2Nosolution–2x+y=3–4x+2y=2Linesareparallel.Nopointofintersection.Nosolutions.SolutionsforsystemoflinearequationsFigure1.1Uniquesolutionx+y=52x-y=4Linesintersectat(3,2)Uniquesolution:x=3,y=2.Figure1.3Manysolution4x–2y=66x–3y=9Bothequationshavethesamegraph.Anypointonthegraphisasolution.Manysolutions.Ch1_3Ch1_4DefinitionAlinearequationinnvariablesx1,x2,x3,…,xnhastheforma1x1+a2x2+a3x3+…+anxn=bwherethecoefficientsa1,a2,a3,…,anandbareconstants.Englishnameofcommonnumeralsystem：naturalnumber,integer,rationalnumber,realnumber,complexnumberpositive,negativeCh1_5Alinearequationinthreevariablescorrespondstoaplaneinthree-dimensionalspace.Uniquesolution※Systemsofthreelinearequationsinthreevariables:Ch1_6NosolutionsManysolutionsCh1_7Howtosolveasystemoflinearequations?Gauss-Jordanelimination.Ch1_8Definition•Amatrixisarectangulararrayofnumbers.•Thenumbersinthearrayarecalledtheelementsofthematrix.Matrices----1298520653C385017B157432ACh1_9SubmatrixAmatrix215032471-ARowandColumn3column2column1column2row1row145372157432----157432--AAofssubmatrice2541137153271-RQPCh1_10IdentityMatricesdiagonalis1，othersare0，ThesubscriptofIissize100010001100132IILocation7,41574322113---aaAaijiselementofrowi,columnjSizeandTypematrixcolumnamatrixrowamatrixsquareamatrix13matrix41matrix3332:Size2385834853109752542301----Ch1_11matrixofcoefficientandaugmentedmatrix623322321321321---xxxxxxxxxRelationsbetweensystemoflinearequationsandmatricestcoefficienofmatrix211132111--matrixaugmented621131322111---Ch1_12ElementaryTransformation1.Interchangetwoequations.2.Multiplybothsidesofanequationbyanonzeroconstant.3.Addamultipleofoneequationtoanotherequation.ElementaryRowOperation1.Interchangetworowsofamatrix.2.Multiplytheelementsofarowbyanonzeroconstant.3.Addamultipleoftheelementsofonerowtothecorrespondingelementsofanotherrow.ElementaryRowOperationsofMatricesCh1_13Example1Solvingthefollowingsystemoflinearequation.623322321321321---xxxxxxxxx---621131322111623322321321321---xxxxxxxxxSolutionEquationMethodInitialsystem:AnalogousMatrixMethodAugmentedmatrix:1232321--xxxxxEq2+(–2)Eq1Eq3+(–1)Eq1-----832011102111R2+(–2)R1R3+(–1)R1rowequivalent83232---xxCh1_1410513233231----xxxxx----0150011103201213233231--xxxxx--210011103201211321-xxx-210010101001Eq1+(–1)Eq2Eq3+(2)Eq2(–1/5)Eq3Eq1+(–2)Eq3Eq2+Eq3Thesolutionis.2,1,1321-xxxThesolutionis.2,1,1321-xxx832123232321-----xxxxxxx-----832011102111R1+(–1)R2R3+(2)R2(–1/5)R3R1+(–2)R3R2+R3Ch1_15Basiccolumnoperationsymboldescription：R1+(–3)R2R1(firstcolumn)plus(-3)R2，SoR1ischanged，R2staysthesameForexample:-----83311851212421(1)R1+2R2-----83311851212421(2)R2+2R1-----833118512481445-----833142135412421Ch1_16Notgood！2R1+R22R1R1+R2Good！…001…001001100010001…Ch1_17Example2Solvingthefollowingsystemoflinearequation.83318521242321321321-----xxxxxxxxxSolution-----83311851212421---4110633012421R231---4110211012421--620021108201--310021108201R1R32)R1(R2-R2)1(R3(2)R2R1-R321310010102001R3R22)R3(R1-.312solution321xxxCh1_18Example3Solvethesystem723286344128421321321-----xxxxxxxx-----701232863441284-----70123286311321---15630110011321---1100521011321--110052101101.110030102001-.1,3,2issolutionThe321-xxxR231R12R33)R1(R2-R3R2---11001563011321R1412)R2(R1-2R3R21)R3(R1-SolutionCh1_19Summary-----701232863441284]:[BAABUserowoperationsto[A:B]:.110030102001------701232863441284]:[]:[XIBAni.e.,Def.[In:X]iscalledthereducedechelonformof[A:B].Note.1.IfAisthematrixofcoefficientsofasystemofnequationsinnvariablesthathasauniquesolution,thenAisrowequivalenttoIn(AIn).2.IfAIn,thenthesystemhasuniquesolution.723286344128421321321-----xxxxxxxxCh1_20Example4ManySystemsSolvingthefollowingthreesystemsoflinearequation,allofwhichhavethesamematrixofcoefficients.33212321132142for423bxxxbxxxbxxx----inturn433,210,11118321--bbbSolution------42114213111412308311.212,130,211321321321--xxxxxxxxx-------123110315210308311---212100315210013101--212100131010201001R2+(–2)R1R3+R11)R2(R3R2R1-R32R2R3)1