LinearAlgebra（2credits）2.1.4TheTransposeofaMatrixandSymmetricMatrixForexample,8,54221A;825241TA6,B18.618TB1.TheTransposeofaMatrixThetransposeofanmatrixisthematrixdefinedbyforandmnAnmjiijba1,2,,jn1,2,,.imThetransposeofisdenotedbyAT.ARulesforTranspose;1AATT;2TTTBABA;3TTAA....
LinearAlgebra（2credits）2.1.3MatrixMultiplication1.DefinitionIfisanmatrixandisanmatrix,thentheproductisanmatrixwhoseentriesaredefinedbyijAa11221sijijijissjikkjkcababababmsijBbsnijCABcmn(1,2,,;1,2,,)imjnEXAMPLEIf,compute415003112101A121113121430BAB.Thus...
LinearAlgebra（2credits）2.1.2AdditionandScalarMultiplicationofMatricesDefinitionmnmnmmmmnnnnbababababababababaBA2211222222212111121211111.MatrixAdditionIfarebothmatrices,thenthesumisanmatrixasmn,ijijAaBbABmnNoteTwomatriceswiththesamedimensionscanbeaddedbyaddingtheircorrespondingentries.Forexample,...
LinearAlgebra（2credits）2.1.1MatrixNotation11112211211222221122nnnnnnnnnnaxaxaxbaxaxaxbaxaxaxb1.SystemsoflinearequationsThesolutionsrelyon,1,2,,,aijijncoefficients1,2,,ibinconstants1.IntroductionofMatrixnnnnnnnbaaabaaabaaa21222221111211Tostudythepropertiesofalinearsys...
3.4.3TransitionmatrixLinearAlgebraDefinitionIfandaretwobasesofthevectorspace,and𝑉.Denote𝑃thetransitionmatrixfrombasistobasis.basistransformationformula(𝛽1,𝛽2,⋯,𝛽𝑛)=(𝛼1,𝛼2,⋯,𝛼𝑛)𝑃ExampleThetransitionmatrixfrombasistobasisofis.(23−1−2)Solution:Giventhetransitionmatrix,𝑃(𝛼1,𝛼2)𝑃=(𝛽1,𝛽2).⇒𝑃=¿¿(110−1)−1(1112)¿(110−1)(1112)¿(23−1−2).ExampleSupposeProofthatisabasisofandf...
3.4.2Basis,DimensionandCoordinatesofVectorSpaceLinearAlgebraDefinitionLetVbeavectorspace，islinearlyindependent；(2)AnyvectorinVcanbelinearlyrepresentedbyThevectorgroupiscalledabasisofthevectorspace,The𝑉numberofvectorsinabasisiscalledthedimensionofthevectorspace,andiscalleda𝑉𝑉r-dimensionalvectorspace.Stipulation:Thedimensionofthezerovectorspaceis0.1)Thebasisofisthelargestindependentgroupof,a...
3.4.1TheDefinitionofVectorSpaceLinearAlgebraDefinition(VectorSpace)Avectorsetiscalledavectorspaceifithas𝑉thefollowingthreeproperties.Vectorsetisnotempty𝑉(closedundervectoraddition)Ifandbelongto,belongstoV𝑉(closedundervectoraddition)Ifbelongstoandisascalar,belongstoV𝑉2)Thedefinitionalsospecifiesstepstoverifywhetheravectorsetisavectorspace:①Visnotempty;②Visclosedundervectoraddition;③...
LinearAlgebra3.3.3CalculationofthelargestIndependentgroupandRankofVectorGroupsExampleCalculatetherankandalargestindependentgroupofthefollowingvectorgroup:Solution:Method1Discriminatingthecorrelationofvectorgroups.𝛼3=𝛼1+2𝛼2.arelinearlydependentandarelinearlyindependent.𝛼1=(32−1−3−2),Thus,therankofthevectorgroupis2,andisalargestindependentgroupTheorem8TherankofmatrixAisequaltotherankofther...
LinearAlgebra3.3.2PropertiesoftheLargestIndependentGroupsandRankQuestion3.Ifthevectorgroupitselfislinearlyindependent,whatisthelargestindependentgroupofthevectorgroup?Property1.AvectorgroupislinearindependentThenumberofvectorsitcontainsisequaltoitsrank.1=1002=,0103=001Conclusion1.I0:1,2,,rislinearlyindependent,thenthelargestindependentgroupofI0isitself.Forexample,in11,01,10,3=1=...
LinearAlgebra3.3.1DefinitionofTheLargestIndependentGroupandRankofVectorGroupsR:255G:255B:0R:255G:0B:255R:0G:255B:2552550RedGreenBlue002552550002550Thepracticalmeaningofn-dimensionalvectorswhereVisiblecolorsinnaturecanbeobtainedbymixingprimarycolorswithcertainproportions,thatis,anycolorisregardedasalinearcombinationofred,greenandbluecolorvectors.Theprimarycolorsshouldbeindependenttoeachother,tha...
LinearAlgebra3.2.2PropertiesofLinearCorrelationofVectorGroupsTheorem6VectorgroupA:islinearlydependent,ÛTherankofthematrixA=()<m(thenumberofvectors).VectorgroupA:islinearlyindependent,ÛTherankofthematrixA=()=m(thenumberofvectors).Note:1)Theconclusionalsoholdsfortherowvectorcase.ExampleDeterminethelinearcorrelationofvectorgroupSolution:LetObviously,usingTheorem6todeterminethecorrelationisverysi...
LinearAlgebra3.2.2PropertiesofLinearCorrelationofVectorGroupsTheorem1VectorgroupA:islinearlydependentThereisatleastoneofthevectorsisalinearcombinationofother−1vectors.𝑚Proof(Sufficiency)Givenavectorin(suchas)thatcanbelinearlyrepresentedbytheremainingvectors,so𝛼𝑚=𝜆1𝛼1+𝜆2𝛼2+⋯+𝜆𝑚−1𝛼𝑚−1⇒𝜆1𝛼1+𝜆2𝛼2+⋯+𝜆𝑚−1𝛼𝑚−1+(−1)𝛼𝑚=𝑂Obviouslyarenotallzero,soislinearlydependent.Thatis...
LinearAlgebra3.2.1DefinitionofLinearDependentofVectorGroupsandarecollinearGeometry:Thereisauniquerealscalark,suchthat=kTherearerealscalarssuchthat.Geometry:𝑘2¿𝑘1+𝑘2andarenotcollinear;and,arecoplanar.TherearerealscalarssuchthatTherearerealscalarssuchthatDefinitionVectorgroupislinearlydependent,iftherearescalarsthatarenotallzero,suchthatOtherwise,itisca...
LinearAlgebra3.1.3LinearRepresentationandEquivalentofVectorGroups1.RelationshipBetweenVectorGroupsVectorgroupislinearlyrepresentedbyvectorgroupifeachvectorinthegroupcanberepresentedbythevectorsinthegroup.𝐵canbelinearlyrepresentedbyForexample,2030,1001,ButB:2030cannotbelinearlyrepresentedby1001,Definition,Proof:GivenSincevectorgroupislinearlyrepresentedbyvectorgroup,𝐴𝐵therearescalarssuchthatC...
LinearAlgebra3.1.2DefinitionandPropertiesofLinearCombinationandLinearRepresentation;andarecollinear.Geometry:Thereisauniquerealscalarksuchthat=kCalled:canbelinearlyrepresentedby.Geometry:k1k2=k1+k2andarenotcollinear;,andarecoplanarTherearerealscalarsk1、k2suchthat=k1+k2Called:canbelinearlyrepresentedby,DefinitionGivenavectorgroup,sca...
LinearAlgebra3.1.1VectorandLinearOperation)HorizontalAngleofFuselage:ElevationAngleofFuselage:AngleoftheWing：Todeterminethestatusoftheaircraft,a6-dimensionalvectorisrequired:Descriptionofaircraftflightstatus¿𝜽(𝟎≤𝜽<𝟐𝝅)𝝍(−𝝅<𝝍≤𝝅)𝝓(−𝝅𝟐≤𝝓≤𝝅𝟐)Todeterminethestatusoftheaircraft,thefollowing6parametersarerequired:Thepositionparametersofthefocusinspace:R:255G:255B:0R:255G:0B:255255...
第2章矩阵主要内容1.高斯消去法2.矩阵的加法、数乘、乘法3.矩阵的转置、对称矩阵4.逆矩阵5.矩阵的初等变换和初等矩阵6.分块矩阵§2.1高斯消元法在第一章，求解n个未知数n个方程的线性方程组的克拉默法则。那么如何求解n个未知元m个方程的线性方程组？(mn）在本节我们进一步介绍在中学里所熟知的高斯消去法。高斯消去法是用代入消元法或加减消元法，化为容易求解的同解方程组。例1用加减消元法解三元一次方程组x12x25x32...
在解析几何中,为了便于研究二次曲线cossin,sincos.xxyyxy把方程化为标准形22.mxnyd的几何性质,我们可以选择适当的坐标旋转变换ax2+bxy+cy2=d第6章二次型6.1二次型的定义和矩阵表示合同矩阵其中系数是数域F中的数，叫做数域F上的n元二次型（简称二次型）。实数域上的二次型简称实二次型。定义6.1n元变量x1,x2,,xn的二次齐次多项式212111121213131122222323222(,,,)22222nnnnnnnnfxxxaxa...
主要内容Rn的基与向量关于基的坐标Rn中向量的内积标准正交基和正交矩阵说明：本章重点是第一节和第二节的内容,第三节至第六节的内容自己阅读.若时间允许，我们再做详细讨论.第4章向量空间和线性变换4.1Rn的基及向量关于基的坐标niijnnnnnn:R0,,0,1,0,,0,1,2,,;nAa,A0,AnnRn1RRnRin从前面的知识我们知道中的单位向量是线性无关的一个阶实矩阵如果则的个线性行向量和个列向量也都是线性无关的.我们...
线性代数模拟测验（7）评分标准一.选择题（本大题共5小题，每小题3分，共15分）BDCDA二、填空题（本大题共5小题，每小题4分，满分20分）6.7.408.9.10.三、计算题（本大题共3小题，每小题8分，满分24分）11.解：--------4分--------8分12.解：原式--------2分--------4分--------6分--------8分13.解：---3分---6分故-------8分四、解答题（本大题共5小题，满分41分）第1页共3页14.解：--------4分因此向量组的秩--------6分向量...
