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 


FINITEDIMENSIONAL 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 FiniteDimensional 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 FINITEDIMENSIONAL SPACES 


A.1. 
Introduction 


A.2. 
The Case of Complete Algebras 


A.3. 
The Case of OneDimensional 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 