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Linear Algebra

Georgi E. Shilov
Publisher: 
Dover Publications
Publication Date: 
1977
Number of Pages: 
387
Format: 
Paperback
Price: 
16.95
ISBN: 
048663518X
Category: 
Textbook
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on
12/31/2010
]

This is a very traditional course in the theory of linear algebra. It includes a detailed study of determinants, matrices, and quadratic forms, mixed with more abstract treatments of general linear spaces over a general field and of real and complex inner product spaces, and ending with chapters on algebras and on categories. Algebras are also used in the discussion of Jordan canonical form. The book is a translation by Richard A. Silverman of a Russian-language work; the prose is clear and easy to follow.

The book consists mostly of a narrative of definitions, theorems, and proofs, but is well-illustrated with many brief examples. There are a reasonable number of exercises, some numeric, but most asking for proofs and intended to continue subjects brought up in the text. The exercises are collected at the end each chapter rather than right after the section they apply to. All exercises have hints, and sometimes answers, in the back of the book.

The book largely ignores the practice of linear algebra. It does not deal at all with computational aspects such as LU-factorization or QR-factorization or even Gaussian elimination. There are no applications given, other than some applications to other areas of mathematics buried in the examples. The applications of linear algebra are so important today that omitting them is a serious drawback, and makes it difficult to motivate the development of the subject.

Bottom line: a competent. concrete, and easy-to-understand course in traditional linear algebra at a bargain price, but limited to the pure-math aspects and focused on concepts and proofs rather than techniques.


Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

 

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

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