An Introduction to Linear Algebra

Rigorous, self-contained coverage of determinants, vectors, matrices and linear equations, quadratic forms, more. Elementary, easily readable account with numerous examples and problems at the end of each chapter. "The straight-forward clarity of the writing is admirable."American Mathematical Monthly. Bibliography.

PART I
"DETERMINANTS, VECTORS, MATRICES, AND LINEAR EQUATIONS"
I. DETERMINANTS
  1.1. Arrangements and the Î-symbol
  1.2. Elementary properties of determinants
  1.3. Multiplication of determinants
  1.4. Expansion theorems
  1.5. Jacobi's theorem
  1.6. Two special theorems on linear equations
II. VECTOR SPACES AND LINEAR MANIFOLDS
  2.1. The algebra of vectors
  2.2. Linear manifolds
  2.3. Linear dependence and bases
  2.4. Vector representation of linear manifolds
  2.5. Inner products and orthonormal bases
III. THE ALGEBRA OF MATRICES
  3.1. Elementary algebra
  3.2. Preliminary notions concerning matrices
  3.3. Addition and multiplication of matrices
  3.4. Application of matrix technique to linear substitutions
  3.5. Adjugate matrices
  3.6. Inverse matrices
  3.7. Rational functions of a square matrix
  3.8. Partitioned matrices
IV. LINEAR OPERATIONS
  4.1. Change of basis in a linear manifold
  4.2. Linear operators and their representations
  4.3. Isomorphisms and automorphisms of linear manifolds
  4.4. Further instances of linear operators
V. SYSTEMS OF LINEAR EQUATIONS AND RANK OF MATRICES
  5.1. Preliminary results
  5.2. The rank theorem
  5.3. The general theory of linear equations
  5.4. Systems of homogeneous linear equations
  5.5. Miscellaneous applications
  5.6. Further theorems on rank of matrices
VI. ELEMENTARY OPERATIONS AND THE CONCEPT OF EQUIVALENCE
  6.1. E-operations and E-matrices
  6.2. Equivalent matrices
  6.3. Applications of the preceding theory
  6.4. Congruence transformations
  6.5. The general concept of equivalence
  6.6. Axiomatic characterization of determinants
PART II
FURTHER DEVELOPMENT OF MATRIX THEORY
VII. THE CHARACTERISTIC EQUATION
  7.1. The coefficients of the characteristic polynomial
  7.2. Characteristic polynomials and similarity transformations
  7.3. Characteristic roots of rational functions of matrices
  7.4. The minimum polynomial and the theorem of Cayley and Hamilton
  7.5. Estimates of characteristic roots
  7.6. Characteristic vectors
VIII. ORTHOGONAL AND UNITARY MATRICES
  8.1. Orthogonal matrices
  8.2. Unitary matrices
  8.3. Rotations in the plane
  8.4. Rotations in space
IX. GROUPS
  9.1. The axioms of group theory
  9.2. Matrix groups and operator groups
  9.3. Representation of groups by matrices
  9.4. Groups of singular matrices
  9.5. Invariant spaces and groups of linear transformations
X. CANONICAL FORMS
  10.1. The idea of a canonical form
  10.2. Diagonal canonical forms under the similarity group
  10.3. Diagonal canonical forms under the orthogonal similarity group and the unitary similarity group
  10.4. Triangular canonical forms
  10.5. An intermediate canonical form
  10.6. Simultaneous similarity transformations
XI. MATRIX ANALYSIS
  11.1 Convergent matrix sequences
  11.2 Power series and matrix functions
  11.3 The relation between matrix functions and matrix polynomials
  11.4 Systems of linear differential equations
PART III
QUADRIATIC FORMS
XII. "BILINEAR, QUADRATIC, AND HERMITIAN FORMS"
  12.1 Operators and forms of the bilinear and quadratic types
  12.2 Orthogonal reduction to diagonal form
  12.3 General reduction to diagonal form
  12.4 The problem of equivalence. Rank and signature
  12.5 Classification of quadrics
  12.6 Hermitian forms
XIII. DEFINITE AND INDEFINITE FORMS
  13.1 The value classes
  13.2 Transformations of positive definite forms
  13.3 Determinantal criteria
  13.4 Simultaneous reduction of two quadratic forms
  13.5 "The inequalities of Hadamard, Minkowski, Fischer, and Oppenheim"
  MISCELLANEOUS PROBLEMS
  BIBLIOGRAPHY
  INDEX