"Linear Algebra" is intended for a one-term course at the junior or senior level. It begins with an exposition of the basic theory of vector spaces and proceeds to explain the fundamental structure theorem for linear maps, including eigenvectors and eigenvalues, quadratic and hermitian forms, diagnolization of symmetric, hermitian, and unitary linear maps and matrices, triangulation, and Jordan canonical form. The book also includes a useful chapter on convex sets and the finite-dimensional Krein-Milman theorem. The presentation is aimed at the student who has already had some exposure to the elementary theory of matrices, determinants and linear maps. However the book is logically self-contained. In this new edition, many parts of the book have been rewritten and reorganized, and new exercises have been added.
Author(s): Serge Lang
Series: Undergraduate Texts in Mathematics
Edition: 3
Publisher: Springer
Year: 2004
Language: English
Pages: 285
Foreword
Contents
I Vector Spaces
§1. DEFINITIONS
§2. BASES
§3. DIMENSION OF A VECTOR SPACE
§4. SUMS AND DIRECT SUMS
II Matrices
§1. THE SPACE OF MATRICES
§2. LINEAR EQUATIONS
§3. MULTIPLICATION OF MATRICES
III Linear Mappings
§1. MAPPINGS
§2. LINEAR MAPPINGS
§3. THE KERNEL AND IMAGE OF A LINEAR MAP
§4. COMPOSITION AND INVERSE OF LINEAR MAPPINGS
§5. GEOMETRIC APPLICATIONS
IV Linear Maps and Matrices
§1. THE LINEAR MAP ASSOCIATED WITH A MATRIX
§2. THE MATRIX ASSOCIATED WITH A LINEAR MAP
§3. BASES, MATRICES, AND LINEAR MAPS
V Scalar Products and Orthogonality
§1. SCALAR PRODUCTS
§2. ORTHOGONAL BASES, POSITIVE DEFINITE CASE
§3. APPLICATION TO LINEAR EQUATIONS; THE RANK
§4. BILINEAR MAPS AND MATRICES
§5. GENERAL ORTHOGONAL BASES
§6. THE DUAL SPACE AND SCALAR PRODUCTS
§7. QUADRATIC FORMS
§8. SYLVESTER'S THEOREM
VI Determinants
§1. DETERMINANTS OF ORDER 2
§2. EXISTENCE OF DETERMINANTS
§3. ADDITIONAL PROPERTIES OF DETERMINANTS
§4. CRAMER'S RULE
§5. TRIANGULATION OF A MATRIX BY COLUMN OPERATIONS
§6. PERMUTATIONS
§7. EXPANSION FORMULA AND UNIQUENESS OF DETERMINANTS
§8. INVERSE OF A MATRIX
§9. THE RANK OF A MATRIX AND SUBDETERMINANTS
VII Symmetric, Hermitian, and Unitary Operators
§1. SYMMETRIC OPERATORS
§2. HERMITIAN OPERATORS
§3. UNITARY OPERATORS
VIII Eigenvectors and Eigenvalues
§1. EIGENVECTORS AND EIGENVALUES
§2. THE CHARACTERISTIC POLYNOMIAL
§3. EIGENVALUES AND EIGENVECTORS OF SYMMETRIC MATRICES
§4. DIAGONALIZATION OF A SYMMETRIC LINEAR MAP
§5. THE HERMITIAN CASE
§6. UNITARY OPERATORS
IX Polynomials and Matrices
§1. POLYNOMIALS
§2. POLYNOMIALS OF MATRICES AND LINEAR MAPS
X Triangulation of Matrices and Linear Maps
§1. EXISTENCE OF TRIANGULATION
§2. THEOREM OF HAMILTON-CAYLEY
§3. DIAGONALIZATION OF UNITARY MAPS
XI Polynomials and Primary Decomposition
§1. THE EUCLIDEAN ALGORITHM
§2. GREATEST COMMON DIVISOR
§3. UNIQUE FACTORIZATION
§4. APPLICATION TO THE DECOMPOSITION OF A VECTOR SPACE
§5. SCHUR'S LEMMA
§6. THE JORDAN NORMAL FORM
XII Convex Sets
§1. DEFINITIONS
§2. SEPARATING HYPERPLANES
§3. EXTREME POINTS AND SUPPORTING HYPERPLANES
§4. THE KREIN-MILMAN THEOREM
APPENDIX I Complex Numbers
APPENDIX II Iwasawa Decomposition and Others
Index
