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Overview

For courses in Advanced Linear Algebra. 


Illustrates the power of linear algebra through practical applications 

This acclaimed theorem-proof text presents a careful treatment of the principal topics of linear algebra. It emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate. Applications to such areas as differential equations, economics, geometry, and physics appear throughout, and can be included at the instructor’s discretion.  


0134860241 / 9780134860244 Linear Algebra, 5/e

Table of contents

* Sections denoted by an asterisk are optional.


1. Vector Spaces 

1.1 Introduction 

1.2 Vector Spaces 

1.3 Subspaces 

1.4 Linear Combinations and Systems of Linear Equations 

1.5 Linear Dependence and Linear Independence 

1.6 Bases and Dimension 

1.7* Maximal Linearly Independent Subsets 

Index of Definitions 


2. Linear Transformations and Matrices 

2.1 Linear Transformations, Null Spaces, and Ranges 

2.2 The Matrix Representation of a Linear Transformation 

2.3 Composition of Linear Transformations and Matrix Multiplication 

2.4 Invertibility and Isomorphisms 

2.5 The Change of Coordinate Matrix

2.6* Dual Spaces 

2.7* Homogeneous Linear Differential Equations with Constant Coefficients 

Index of Definitions 


3. Elementary Matrix Operations and Systems of Linear Equations

3.1 Elementary Matrix Operations and Elementary Matrices 

3.2 The Rank of a Matrix and Matrix Inverses 

3.3 Systems of Linear Equations – Theoretical Aspects 

3.4 Systems of Linear Equations – Computational Aspects 

Index of Definitions


4. Determinants

4.1 Determinants of Order 2

4.2 Determinants of Order n 

4.3 Properties of Determinants 

4.4 Summary|Important Facts about Determinants 

4.5* A Characterization of the Determinant 

Index of Definitions


5. Diagonalization

5.1 Eigenvalues and Eigenvectors 

5.2 Diagonalizability 

5.3* Matrix Limits and Markov Chains 

5.4 Invariant Subspaces and the Cayley–Hamilton Theorem 

Index of Definitions


6. Inner Product Spaces

6.1 Inner Products and Norms 

6.2 The Gram–Schmidt Orthogonalization Process and Orthogonal Complements 

6.3 The Adjoint of a Linear Operator

6.4 Normal and Self-Adjoint Operators 

6.5 Unitary and Orthogonal Operators and Their Matrices

6.6 Orthogonal Projections and the Spectral Theorem

6.7* The Singular Value Decomposition and the Pseudoinverse 

6.8* Bilinear and Quadratic Forms 

6.9* Einstein's Special Theory of Relativity 

6.10* Conditioning and the Rayleigh Quotient 

6.11* The Geometry of Orthogonal Operators

Index of Definitions


7. Canonical Forms

7.1 The Jordan Canonical Form I 

7.2 The Jordan Canonical Form II

7.3 The Minimal Polynomial 

7.4* The Rational Canonical Form 

Index of Definitions


Appendices

A. Sets 

B. Functions 

C. Fields 

D. Complex Numbers 

E. Polynomials 


Answers to Selected Exercises

Index

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