MATH 7101 – Advanced Linear Algebra

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<ul><li><p>MATH7101AdvancedLinearAlgebraFall2016</p><p>Instructor AmerIqbalEmail iqbal.amer@gmail.comamer@alum.mit.edu</p><p>CourseBasicsCreditHours 3Lectures NumberofLecturesinaWeek 2 Duration 90minTutorials NumberofTutorialsinaWeek</p><p>(afterfirst4lectures)0 Duration </p><p>COURSEDESCRIPTIONThis course builds on the undergraduate linear algebra course. The topics covered will be: Vector Spaces, Subspaces, Direct Sums, Relation with Matrices, Linear Transformations, Kernel and Image of a Linear Transformation, TheRank PlusNullity Theorem, Linear transformation and matrices, Invariant Subspaces, Quotient Spaces, Isomorphism Theorems, Dual Basis, Operator Adjoints, Eigenvalues and eigenvectors, Characteristic polynomial, Diagonal form, Jordan canonical form, BilinearMaps, Tensor Products, Linear Transformations on a Tensor Product, Tensor Product of Linear Transformations, The Symmetric andAntisymmetric tensor products. Applicationstovariousproblemsofabovementionedtopicswillbeprovidedaswell.</p><p>COURSEPREREQUISITE(S)</p><p>Undergraduatelinearalgebracourse</p><p>COURSEOBJECTIVESThiscourseisaboutlinearvectorspaceandvariousstructuresdefinedonthem.Itshouldbethoughtofasacontinuationoftheundergraduatelinearalgebracoursebutwithmoreemphasisonabstractreasoningratherthanmatrixcalculations.Thiscoursewillexplaintostudentstherelationbetweenvectorspacesandmatrices,theuseoflineartransformationstostudyvariouspropertiesoftheunderlyingvectorspaces,variousisomorphismtheoremsassociatedwithlineartransformationsandhownewvectorsspacescanbeconstructedusingtheideaoftensorproducts.</p><p>LearningOutcomesAttheendofthecoursestudentsshouldbeabletounderstandthebasicfeaturesofavectorspaceandlineartransformationsdefinedonthemandunderstandtheisomorphismtheorems.TheyshouldbeabletounderstandthedecompositionofthevectorspaceintovariousinvariantsubspacesandtherelationshipofthisdecompositionwiththeJordancanonicalform.Theyshouldalsounderstandthetensorproductofvectorsspacesanddetermineitsbasisandthecorrespondingmatrices.HomeWork:10%ClassPresentation:20%MidTerm:20%FinalExam:50%</p><p>mailto:iqbal.amer@gmail.com</p></li><li><p>ExaminationDetail</p><p>MidtermExam</p><p>Acomprehensiveexaminationof2hours</p><p>FinalExam</p><p>Acomprehensiveexaminationof3hours</p><p>COURSEOVERVIEWWeek Topics Particulars</p><p>12 ReviewofMatrixAlgebraandVectorSpaces VectorSpaces,Basis,Matrices,BasisChange,LinearTransformations,MatrixofLT,DualSpaces,Basis,Subspaces,DirectSums,SpanningSetsandLinearIndependence,TheDimensionofaVectorSpace,TheComplexificationofaRealVectorSpace</p><p>25 LinearTransformations LinearTransformations,KernelandImageofLT,Isomorphisms,TheRankPlusNullityTheorem,ChangeofBasesforLinearTransformations,InvariantSubspacesandReducingPairs</p><p>68 IsomorphismTheorems(Chapter3ofthetextbook) QuotientSpaces,UniversalPropertyofQuotients,FirstIsomorphismTheorem,Complements,LinearFunctionals,DualBases,OperatorAdjoints</p><p>78 Eigenvalues&amp;Eigenvectors TheCharacteristicPolynomial,TheRationalCanonicalForm,GeometricandAlgebraicMultiplicities,JordanCanonicalForm,Galoisgroupofapolynomial.</p><p>910 InnerProductSpaces Norm,Isometries,Orthogonality,OrthogonalandOrthonormalSets,TheProjectionTheorem,TheRieszRepresentationTheorem</p><p>1112 BilinearForms Symmetric,SkewSymmetricandAlternateForms,TheMatrixofaBilinearForm,OrthogonalProjections,LinearFunctionals,OrthogonalComplementsandOrthogonalDirectSums,Isometries,HyperbolicSpaces</p><p>1314 Tensors BilinearMaps,TensorProducts,DefiningLinearTransformationsonaTensorProduct,TheTensorProductofLinearTransformations,MultilinearMapsandIteratedTensorProducts</p><p>Textbooks/SupplementaryReading</p><p>1.StevenRoman,AdvancedLinearAlgebra,PublishedbySpringer(2008).2.JonathanGolan,TheLinearAlgebraaBeginningGraduateStudentOughttoKnow,PublishedbySpringer(2007).</p></li></ul>