This top-selling, theorem-proof book presents a careful treatment of the principle topics of linear algebra, and illustrates the power of the subject through a variety of applications. It emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate. Chapter topics cover vector spaces, linear transformations and matrices, elementary matrix operations and systems of linear equations, determinants, diagonalization, inner product spaces, and canonical forms.
Author(s): Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence
Edition: 4th
Publisher: Prentice Hall
Year: 2003
Language: English
Pages: 615
Contents......Page f005.djvu
Preface......Page f009.djvu
1.1 Introduction......Page p001.djvu
1.2 Vector Spaces......Page p006.djvu
1.3 Subspaces......Page p016.djvu
1.4 Linear Combinations and Systems of Linear Equations......Page p024.djvu
1.5 Linear Dependence and Linear Independence......Page p035.djvu
1.6 Bases and Dimension......Page p042.djvu
1.7* Maximal Linearly Independent Subsets......Page p058.djvu
Index of Definitions......Page p062.djvu
2.1 Linear Transformations, Null Spaces, and Ranges......Page p064.djvu
2.2 The Matrix Representation of a Linear Transformation......Page p079.djvu
2.3 Composition of Linear Transformations and Matrix Multiplication......Page p086.djvu
2.4 Invertibility and Isomorphisms......Page p099.djvu
2.5 The Change of Coordinate Matrix......Page p110.djvu
2.6* Dual Spaces......Page p119.djvu
2.7* Homogeneous Linear Differential Equations with Constant Coefficients......Page p127.djvu
Index of Definitions......Page p145.djvu
3.1 Elementary Matrix Operations and Elementary Matrices......Page p147.djvu
3.2 The Rank of a Matrix and Matrix Inverses......Page p152.djvu
3.3 Systems of Linear Equations - Theoretical Aspects......Page p168.djvu
3.4 Systems of Linear Equations - Computational Aspects......Page p182.djvu
Index of Definitions......Page p198.djvu
4.1 Determinants of Order 2......Page p199.djvu
4.2 Determinants of Order n......Page p209.djvu
4.3 Properties of Determinants......Page p222.djvu
4.4 Summary - Important Facts about Determinants......Page p232.djvu
4.5* A Characterization of the Determinant......Page p238.djvu
Index of Definitions......Page p244.djvu
5.1 Eigenvalues and Eigenvectors......Page p245.djvu
5.2 Diagonalizability......Page p261.djvu
5.3* Matrix Limits and Markov Chains......Page p283.djvu
5.4 Invariant Subspaces and the Cayley-Hamilton Theorem......Page p313.djvu
Index of Definitions......Page p328.djvu
6.1 Timer Products and Norms......Page p329.djvu
6.2 The Gram-Schmidt Orthogonalization Process and Orthogonal Complements......Page p341.djvu
6.3 The Adjoint of a Linear Operator......Page p357.djvu
6.4 Normal and Self-Adjoint Operators......Page p369.djvu
6.5 Unitary and Orthogonal Operators and Their Matrices......Page p379.djvu
6.6 Orthogonal Projections and the Spectral Theorem......Page p398.djvu
6.7* The Singular Value Decomposition and the Pseudoinverse......Page p405.djvu
6.8* Bilinear and Quadratic Forms......Page p422.djvu
6.9* Einstein's Special Theory of Relativity......Page p451.djvu
6.10* Conditioning and the Rayleigh Quotient......Page p464.djvu
6.11* The Geometry of Orthogonal Operators......Page p472.djvu
Index of Definitions......Page p480.djvu
7.1 The Jordan Canonical Form I......Page p482.djvu
7.2 The Jordan Canonical Form II......Page p497.djvu
7.3 The Minimal Polynomial......Page p516.djvu
7.4* The Rational Canonical Form......Page p524.djvu
Index of Definitions......Page p548.djvu
A Sets......Page p549.djvu
B Functions......Page p551.djvu
C Fields......Page p552.djvu
D Complex Numbers......Page p556.djvu
E Polynomials......Page p562.djvu
Answers to Selected Exercises......Page p571.djvu
Index......Page p589.djvu
