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1、Annexe3RECOMMENDEDKNOWLEDGEINMATHEMATICS1-ALGEBRA1.1SettheoryOperationsonsets,characteristicfunctions.Maps,injectivity,surjectivity.Directandinverseimageofaset.Integernumbers,finitesets,countability.1.2NumbersandusualstructuresCompositionlaws;groups,rings,fields.Equivalencerelati
2、ons,quotientstructures.Realnumbers,complexnumbers,complexexponential.Applicationtoplanegeometry.Polynomials,relationsbetweentherootsandthecoefficients.Elementaryarithmetics(inZ/nZ).1.3Finitedimensionalvectorspaces(*)Freefamilies,generatingfamilies,bases,dimension.Determinantofnve
3、ctors;characterizationofbases.Matrices,operationsonmatrices.Determinantofasquarematrix;expansionwithrespecttoalineortoacolumn;rank,cofactors.Linearmaps,matrixassociatedtoalinearmap.Endomorphisms,trace,determinant,rank.Linearsystemsofequations.1.4ReductionofendomorphismsStablesubs
4、paces.Eigenvalues,eigenvectorsofanendomorphismorasquarematrix;similarmatrices;geometricalinterpretation.Characteristicpolynomial,Cayley-Hamiltontheorem.Reductionofendomorphismsinfinitedimension;diagonalizableendomorphismsandmatrices.1.5Euclideanspaces,EuclideangeometryScalarprodu
5、ct;Cauchy-Schwarzinequality;normsandassociateddistances.Euclideanspacesoffinitedimension,orthonormalbases;orthogonalprojections.OrthogonalgroupO(E);orthogonalsymmetries.Orthogonalmatrices;diagonalizationofsymmetricrealmatrices.PropertiesoforthogonalendomorphismsofR²andR³.kk(*)Ins
6、everalcountrieslinearalgebraisstudiedonlyinRorC;thecandidatesfromthesecountriesarestronglyadvisedtogetfamiliarwiththeformalismofabstractvectorspaces.2-ANALYSISANDDIFFERENTIALGEOMETRY2.1TopologyinfinitedimensionalnormedvectorspacesOpenandclosedsets,accumulationpoints,interiorpoint
7、s.Convergentsequencesinnormedvectorspaces;continuousmappings.Compactspaces,imagesofcompactsetsbycontinuousmappings,existenceofextrema.Equivalenceofnorms.2.2RealorcomplexvaluedfunctionsdefinedonanintervalkDerivativeatapoint,functionsofclassC.Meanvaluetheorem,Taylor'sformula.Primit
8、iveofcontinuousfunctions.Usualfunctions(
