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Index
Title Page
Copyright Page
Preface
Contents
CHAPTER 1 Vector Spaces
1.1 Introduction
1.2 Scalars
1.3 Cartesian space
1.4 Vector spaces
1.5 Independence of vectors
1.6 Dimension and basis
1.7 Isomorphism
1.8 Subspaces
1.9 Sums and intersections of subspaces
1.10 Subspaces
CHAPTER 2 Linear Transformations and Matrices
2.1 Linear transformations
2.2 Matrix of a linear transformation
2.3 Examples of linear transformations
2.4 Multiplication of linear transformations
2.5 Multiplication of matrices
2.6 Sums and products by scalars
2.7 Sets of linear transformations of a vector space
2.8 Submatrices, partitioned matrices
2.9 Row and column matrices
CHAPTER 3 Systems of Linear Equations
3.1 Rank; row and column spaces of a matrix
3.2 Systems of linear homogeneous equations
3.3 Elementary transformations
3.4 Row-echelon matrices
3.5 Triangular and Hermite matrices
3.6 Properties of Hermite matrices
3.7 Elementary matrices
3.8 Equivalence of matrices
3.9 Nonhomogeneous linear equations
CHAPTER 4 Determinants
4.1 Definition of determinant
4.2 Basic properties of determinants
4.3 Proofs of the properties of determinants
4.4 Classical definition of determinant; Laplace expansion by minors
4.5 Determinants of products of square matrices; determinantal criterion for rank
4.6 Determinants of products of rectangular matrices
4.7 Adjoints and inverses of square matrices; Cramer’s Rule
CHAPTER 5 Equivalence Relations and Canonical Forms
5.1 Equivalence relations
5.2 Canonical forms and invariants
5.3 Alias and alibi
5.4 Change of basis
5.5 Similarity of matrices; eigenvalues
5.6 Equivalence and similarity of linear transformations
CHAPTER 6 Functions of Vectors
6.1 Bilinear forms
6.2 Canonical forms for skew-symmetric and symmetric matrices
6.3 Quadratic forms
6.4 Bilinear functions; dual spaces
6.5 Bilinear scalar functions and matrices; dualities
6.6 Quadratic functions
6.7 Hermitian functions
6.8 Determinants as multilinear functions
CHAPTER 7 Orthogonal and Unitary Equivalence
7.1 Euclidean space and inner products
7.2 Schwartz’s inequality, distance, and angle
7.3 Orthogonality
7.4 Orthogonal subspaces
7.5 Orthogonal transformations
7.6 Diagonalization of symmetric matrices
7.7 Unitary transformations
7.8 Adjoint of a linear transformation
CHAPTER 8 Structure of Polynomial Rings
8.1 Rings and subrings
8.2 Existence and uniqueness of transcendental extensions
8.3 Division algorithm; greatest common divisor
8.4 Factorization of polynomials
8.5 Algebraic extensions of a field
8.6 Congruence of polynomials
8.7 Direct sums of vector spaces
8.8 Idempotents and direct sums
8.9 Decomposition of algebras
8.10 Decomposition of simple algebraic extensions
CHAPTER 9 Equivalence of Matrices over a Ring
9.1 Matrices over a ring
9.2 Equivalence of matrices with polynomial elements
9.3 Matrices with integer elements
9.4 Vector spaces over the integers
9.5 Finitely generated vector spaces over the integers
9.6 Systems of linear differential equations with constant coefficients
CHAPTER 10 Similarity of Matrices
10.1 Minimum function
10.2 Invariant subspaces
10.3 Cayley-Hamilton Theorem
10.4 Primary linear transformations
10.5 Similarity, general case
10.6 Segre characteristics
10.6 Segre characteristics
10.8 Pairs of quadratic forms
10.9 Applications to projective geometry
10.10 Roots of matrices
CHAPTER 11 Linear Inequalities
11.1 Definitions and notation
11.2 Inequalities and convex sets
11.3 Convex cones
11.4 Polar cones and double description
11.5 Linear programming
11.6 The Minimax Theorem
11.7 Matrix games
Appendix I Mathematical Induction
Appendix II Relations and Mappings
Appendix III Bibliography
Glossary of Special Symbols
Index
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