Chapter 1. Linear Equations 1.1. Fields 1.2. Systems of Linear Equations 1.3. Matrices and Elementary Row Operations 1.4. Row-Reduced Echelon Matrices 1.5. Matrix Multiplication 1.6. Invertible Matrices Chapter 2. Vector Spaces 2.1. Vector Spaces 2.2. Subspaces 2.3. Bases and Dimension 2.4. Coordinates 2.5. Summary of Row-Equivalence 2.6. Computations Concerning Subspaces Chapter 3. Linear Transformations 3.1. Linear Transformations 3.2. The Algebra of Linear Transformations 3.3. Isomorphism 3.4. Representation of Transformations by Matrices 3.5. Linear Functionals 3.6. The Double Dual 3.7. The Transpose of a Linear Transformation Chapter 4. Polynomials 4.1. Algebras 4.2. The Algebra of Polynomials 4.3. Lagrange Interpolation 4.4. Polynomial Ideals 4.5. The Prime Factorization of a Polynomial Chapter 5. Determinants 5.1. Commutative Rings 5.2. Determinant Functions 5.3. Permutations and the Uniqueness of Determinants 5.4. Additional Properties of Determinants 5.5. Modules 5.6. Multilinear Functions 5.7. The Grassman Ring Chapter 6. Elementary Canonical Forms 6.1. Introduction 6.2. Characteristic Values 6.3. .Annihilating Polynomials 6.4. Invariant Subspaces 6.5. simultaneous Triangulation; Simultaneous Diagonalization 6.6. Direct-Sum Decompositions 6.7. Invarlant Direct Sums 6.8. The Primary Decomposition Theorem Chapter 7. The Rational and Jordan Forms 7.1. Cyclic Subspaces and Annihilators 7.2. Cyclic Decompositions and the Rational Form 7.3. The Jordan Form 7.4. Computation of Invarlant Factors 7.5. Summary; Semi-Simple Operators Chapter 8. Inner Product Spaces 8.1. Inner Products 8.2. Inner Product Spaces 8.3. Linear Functional] and Adjoints 8.4. Unitary Operators 8.5. Normal Operators Chapter 9. Operators on Inner Product Spaces 9.1. Introduction 9.2. Forms on Inner Product Spaces 9.3. Positive Forms 9.4. More on Forms 9.5. Spectral Theory 9.6. Further Properties of Normal Operators Chapter 10. Bilinear Forms 10.1. Bilinear Forms 10.2. Symmetric Bilinear Forms 10.3. Skew-SymmetricBilinear Forms 10.4 Groups Preserving Bilinear Forms Appendix A.1. Sets A.2. Functions A.3. Equivalence Relations A.4. Quotient Spaces A.5. Equivalence Relations in Linear Algebra A.6. The Axiom of Choice Bibliography lndes
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