Advanced Linear Algebra

Vector SpacesFieldsThe Space FnVector Spaces over an Arbitrary Field Subspaces of Vector SpacesSpan and IndependenceBases and Finite Dimensional Vector SpacesBases and Infinite Dimensional Vector SpacesCoordinate Vectors Linear TransformationsIntroduction to Linear TransformationsThe Range and Kernel of a Linear TransformationThe Correspondence and Isomorphism TheoremsMatrix of a Linear TransformationThe Algebra of L(V, W) and Mmn(F)Invertible Transformations and Matrices PolynomialsThe Algebra of PolynomialsRoots of Polynomials Theory of a Single Linear OperatorInvariant Subspaces of an Operator Cyclic OperatorsMaximal VectorsIndecomposable Linear OperatorsInvariant Factors and Elementary Divisors Canonical FormsOperators on Real and Complex Vector Spaces Inner Product SpacesInner ProductsGeometry in Inner Product SpacesOrthonormal Sets and the Gram-Schmidt Process Orthogonal Complements and ProjectionsDual SpacesAdjoints Linear Operators on Inner Product SpacesSelf-Adjoint and Normal OperatorsSpectral TheoremsNormal Operators on Real Inner Product SpacesUnitary and Orthogonal OperatorsPolar Decomposition and Singular Value Decomposition Trace and Determinant of a Linear OperatorTrace of a Linear OperatorDeterminant of a Linear Operator and MatrixUniqueness of the Determinant of a Linear Operator Bilinear Maps and FormsBasic Properties of Bilinear MapsSymplectic SpacesQuadratic Forms and Orthogonal SpaceReal Quadratic Forms Tensor ProductsIntroduction to Tensor ProductsProperties of Tensor ProductsThe Tensor AlgebraThe Symmetric and Exterior Algebras Appendix A: Answers to Selected ExercisesAppendix B: Hints to Selected Problems Index.