Linear algebra 线性代数

In this volume in his exceptional series of translations of Russian mathematical texts, Richard Silverman has taken Shilov's course in linear algebra and has made it even more accessible and more useful for English language readers.
Georgi E. Shilov, Professor of Mathematics at the Moscow State University, covers determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate trans-formations, 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 algebras and their representations, with an appendix on categories
of finite-dimensional spaces.
The author begins with elementary material and goes easily into the advanced areas, covering all the standard topics of an advanced undergraduate or beginning graduate course. The material is pre-sented in a consistently clear style. Problems are included, with a full section of hints and answers in the back.
Keeping in mind the unity of algebra, geometry and analysis in his approach, and writing practically for the student who needs to learn techniques, Professor Shilov has produced one of the best expositions on the subject. Because it contains an abundance of problems and examples the book will be useful for self-study as well as for the classroom.

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 Compatibility 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
chapter 6 THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR
chapter 7 BILINEAR AND QUADRATIC FORMS
chapter 8 EUCLIDEAN SPACES
chapter 9 UNITARY SPACES
chapter 10 QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES
chapter 11 FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS
Appendix CATEGORIES OF FINITE-DIMENSIONAL SPACES
HINTS AND ANSWERS
BIBLIOGRAPHY
INDEX