Chapter 1. Linear Equations
- 1.1 Fields (no exercises)
- 1.2 Systems of Linear Equations
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) - 1.3 Matrices and Elementary Row Operations
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) - 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 (no exercises)
- 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 (no exercises)
- 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 (no exercises)
- 5.2 Determinant Functions
- 5.3 Permutations and the Uniqueness of Determinants
- 5.4 Additional Properties of Determinants
- 5.5 Modules (no exercises)
- 5.6 Multilinear Functions (no exercises)
- 5.7 The Grassman Ring (no exercises)
Chapter 6. Elementary Canonical Forms
- 6.1 Introduction (no exercises)
- 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 Invariant 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 Invariant Factors
- 7.5 Summary; Semi-simple Operators
Chapter 8. Inner Product Spaces
- 8.1 Inner Products
- 8.2 Inner Product Spaces
- 8.3 Linear Functionals and Adjoints
- 8.4 Unitary Operators
- 8.5 Normal Operators
Chapter 9. Operators on Inner Product Spaces
- 9.1 Introduction (no exercises)
- 9.2 Forms on Inner Product Spaces
(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) - 9.3 Positive Forms
- 9.4 More on Forms (no exercises)
- 9.5 Spectral Theory
- 9.6 Further Properties of Normal Operators (no exercises)
Chapter 10. Bilinear Forms
- 10.1 Bilinear Forms
- 10.2 Symmetric Bilinear Forms
- 10.3 Skew-Symmetric Bilinear Forms
- 10.4 Groups Preserveing Bilinear Forms
