This course was designed for mathematics majors at the junior level, although threefourths of the students were drawn from other scientific and technological disciplines and ranged from freshmen through graduate students. This description of the M.LT. audience for the text remains generally accurate today.

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. Yector 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 Functionais

 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. Multflinear 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. Invariant Direct Sums

 6.8. The Primary Decomposition Theorem

Chapter 7. The Rational and Jordan Forms

 7.1. Cyclic Subspaces and Anuihiators

 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

 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. Billnear Forms

 10.1. Bilinear Forms

 10.2. Symmetric Bilinear Forms

 10.8. 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 Relstions in Linear Algebra

 A.6. The Axiom of Choice

Bibliography

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