Linear Algebra

Determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, Euclidean spaces, unitary spaces, quadratic forms in Euclidean and unitary spaces, finite-dimensional space. Problems; hints and answers.

chapter 1
  DETERMINANTS
  1.1. Number Fields
  1.2. Problems of the Theory of Systems of Linear Equations
  1.3. Determinants of Order n
  1.4. Properties of Determinants
  1.5. Cofactors and Minors
  1.6. Practical Evaluation of Determinants
  1.7. Cramer's Rule
  1.8. Minors of Arbitrary Order. Laplace's Theorem
  1.9. Linear Dependence between Columns
    Problems
chapter 2
  LINEAR SPACES
  2.1. Definitions
  2.2. Linear Dependence
  2.3. "Bases, Components, Dimension"
  2.4. Subspaces
  2.5. Linear Manifolds
  2.6. Hyperplanes
  2.7. Morphisms of Linear Spaces
    Problems
chapter 3
  SYSTEMS OF LINEAR EQUATIONS
  3.1. More on the Rank of a Matrix
  3.2. Nontrivial Compatibility of a Homogeneous Linear System
  3.3. The Compatability Condition for a General Linear System
  3.4. The General Solution of a Linear System
  3.5. Geometric Properties of the Solution Space
  3.6. Methods for Calculating the Rank of a Matrix
    Problems
chapter 4
  LINEAR FUNCTIONS OF A VECTOR ARGUMENT
  4.1. Linear Forms
  4.2. Linear Operators
  4.3. Sums and Products of Linear Operators
  4.4. Corresponding Operations on Matrices
  4.5. Further Properties of Matrix Multiplication
  4.6. The Range and Null Space of a Linear Operator
  4.7. Linear Operators Mapping a Space Kn into Itself
  4.8. Invariant Subspaces
  4.9. Eigenvectors and Eigenvalues
    Problems
chapter 5
  COORDINATE TRANSFORMATIONS
  5.1. Transformation to a New Basis
  5.2. Consecutive Transformations
  5.3. Transformation of the Components of a Vector
  5.4. Transformation of the Coefficients of a Linear Form
  5.5. Transformation of the Matrix of a Linear Operator
  *5.6. Tensors
    Problems
chapter 6
  THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR
  6.1. Canonical Form of the Matrix of a Nilpotent Operator
  6.2. Algebras. The Algebra of Polynomials
  6.3. Canonical Form of the Matrix of an Arbitrary Operator
  6.4. Elementary Divisors
  6.5. Further Implications
  6.6. The Real Jordan Canonical Form
  *6.7. "Spectra, Jets and Polynomials"
  *6.8. Operator Functions and Their Matrices
    Problems
chapter 7
  BILINEAR AND QUADRATIC FORMS
  7.1. Bilinear Forms
  7.2. Quadratic Forms
  7.3. Reduction of a Quadratic Form to Canonical Form
  7.4. The Canonical Basis of a Bilinear Form
  7.5. Construction of a Canonical Basis by Jacobi's Method
  7.6. Adjoint Linear Operators
  7.7. Isomorphism of Spaces Equipped with a Bilinear Form
  *7.8. Multilinear Forms
  7.9. Bilinear and Quadratic Forms in a Real Space
    Problems
chapter 8
  EUCLIDEAN SPACES
  8.1. Introduction
  8.2. Definition of a Euclidean Space
  8.3. Basic Metric Concepts
  8.4. Orthogonal Bases
  8.5. Perpendiculars
  8.6. The Orthogonalization Theorem
  8.7. The Gram Determinant
  8.8. Incompatible Systems and the Method of Least Squares
  8.9. Adjoint Operators and Isometry
    Problems
chapter 9
  UNITARY SPACES
  9.1. Hermitian Forms
  9.2. The Scalar Product in a Complex Space
  9.3. Normal Operators
  9.4. Applications to Operator Theory in Euclidean Space
    Problems
chapter 10
  QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES
  10.1. Basic Theorem on Quadratic Forms in a Euclidean Space
  10.2. Extremal Properties of a Quadratic Form
  10.3 Simultaneous Reduction of Two Quadratic Forms
  10.4. Reduction of the General Equation of a Quadratic Surface
  10.5. Geometric Properties of a Quadratic Surface
  *10.6. Analysis of a Quadric Surface from Its Genearl Equation
  10.7. Hermitian Quadratic Forms
    Problems
chapter 11
  FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS
  11.1. More on Algebras
  11.2. Representations of Abstract Algebras
  11.3. Irreducible Representations and Schur's Lemma
  11.4. Basic Types of Finite-Dimensional Algebras
  11.5. The Left Regular Representation of a Simple Algebra
  11.6. Structure of Simple Algebras
  11.7. Structure of Semisimple Algebras
  11.8. Representations of Simple and Semisimple Algebras
  11.9. Some Further Results
    Problems
*Appendix
  CATEGORIES OF FINITE-DIMENSIONAL SPACES
  A.1. Introduction
  A.2. The Case of Complete Algebras
  A.3. The Case of One-Dimensional Algebras
  A.4. The Case of Simple Algebras
  A.5. The Case of Complete Algebras of Diagonal Matrices
  A.6. Categories and Direct Sums
HINTS AND ANSWERS
BIBLIOGRAPHY
INDEX