, About The Author
Howard Anton obtained his B.A. from Lehigh University, his M.A. from the University of Illinois, and his
Ph.D. from the Polytechnic University of Brooklyn, all in mathematics. In the early 1960s he worked for
Burroughs Corporation and Avco Corporation at Cape Canaveral, Florida, where he was involved with the
manned space program. In 1968 he joined the Mathematics Department at Drexel University, where he taught
full time until 1983. Since then he has devoted the majority of his time to textbook writing and activities for
mathematical associations. Dr. Anton was president of the EPADEL Section of the Mathematical Association
of America (MAA), served on the Board of Governors of that organization, and guided the creation of the
Student Chapters of the MAA. In addition to various pedagogical articles, he has published numerous
research papers in functional analysis, approximation theory, and topology. He is best known for his textbooks
in mathematics, which are among the most widely used in the world. There are currently more than 150
versions of his books, including translations into Spanish, Arabic, Portuguese, Italian, Indonesian, French,
Japanese, Chinese, Hebrew, and German. For relaxation, Dr. Anton enjoys travel and photography.
Copyright © 2010 John Wiley & Sons, Inc. All rights reserved.
, Preface
This edition of Elementary Linear Algebra gives an introductory treatment of linear algebra that is suitable for
a first undergraduate course. Its aim is to present the fundamentals of linear algebra in the clearest possible
way—sound pedagogy is the main consideration. Although calculus is not a prerequisite, there is some
optional material that is clearly marked for students with a calculus background. If desired, that material can
be omitted without loss of continuity.
Technology is not required to use this text, but for instructors who would like to use MATLAB, Mathematica,
Maple, or calculators with linear algebra capabilities, we have posted some supporting material that can be
accessed at either of the following Web sites:
www.howardanton.com
www.wiley.com/college/anton
Summary of Changes in this Edition
This edition is a major revision of its predecessor. In addition to including some new material, some of the old
material has been streamlined to ensure that the major topics can all be covered in a standard course. These
are the most significant changes:
• Vectors in 2-space, 3-space, and n-space Chapters 3 and 4 of the previous edition have been combined
into a single chapter. This has enabled us to eliminate some duplicate exposition and to juxtapose concepts
in n-space with those in 2-space and 3-space, thereby conveying more clearly how n-space ideas generalize
those already familiar to the student.
• New Pedagogical Elements Each section now ends with a Concept Review and a Skills mastery that
provide the student a convenient reference to the main ideas in that section.
• New Exercises Many new exercises have been added, including a set of True/False exercises at the end of
most sections.
• Earlier Coverage of Eigenvalues and Eigenvectors The chapter on eigenvalues and eigenvectors, which
was Chapter 7 in the previous edition, is Chapter 5 in this edition.
• Complex Vector Spaces The chapter entitled Complex Vector Spaces in the previous edition has been
completely revised. The most important ideas are now covered in Section 5.3 and Section 7.5 in the context
of matrix diagonalization. A brief review of complex numbers is included in the Appendix.
• Quadratic Forms This material has been extensively rewritten to focus more precisely on the most
important ideas.
• New Chapter on Numerical Methods In the previous edition an assortment of topics appeared in the last
chapter. That chapter has been replaced by a new chapter that focuses exclusively on numerical methods of
linear algebra. We achieved this by moving those topics not concerned with numerical methods elsewhere
in the text.
• Singular-Value Decomposition In recognition of its growing importance, a new section on Singular-Value
Decomposition has been added to the chapter on numerical methods.
, • Internet Search and the Power Method A new section on the Power Method and its application to
Internet search engines has been added to the chapter on numerical methods.
• Applications There is an expanded version of this text by Howard Anton and Chris Rorres entitled
Elementary Linear Algebra: Applications Version, 10th (ISBN 9780470432051), whose purpose is to
supplement this version with an extensive body of applications. However, to accommodate instructors who
asked us to include some applications in this version of the text, we have done so. These are generally less
detailed than those appearing in the Anton/Rorres text and can be omitted without loss of continuity.
Hallmark Features
• Relationships Among Concepts One of our main pedagogical goals is to convey to the student that linear
algebra is a cohesive subject and not simply a collection of isolated definitions and techniques. One way in
which we do this is by using a crescendo of Equivalent Statements theorems that continually revisit
relationships among systems of equations, matrices, determinants, vectors, linear transformations, and
eigenvalues. To get a general sense of how we use this technique see Theorems 1.5.3, 1.6.4, 2.3.8, 4.8.10,
4.10.4 and then Theorem 5.1.6, for example.
• Smooth Transition to Abstraction Because the transition from Rn to general vector spaces is difficult for
many students, considerable effort is devoted to explaining the purpose of abstraction and helping the
student to “visualize” abstract ideas by drawing analogies to familiar geometric ideas.
• Mathematical Precision When reasonable, we try to be mathematically precise. In keeping with the level
of student audience, proofs are presented in a patient style that is tailored for beginners. There is a brief
section in the Appendix on how to read proof statements, and there are various exercises in which students
are guided through the steps of a proof and asked for justification.
• Suitability for a Diverse Audience This text is designed to serve the needs of students in engineering,
computer science, biology, physics, business, and economics as well as those majoring in mathematics.
• Historical Notes To give the students a sense of mathematical history and to convey that real people
created the mathematical theorems and equations they are studying, we have included numerous Historical
Notes that put the topic being studied in historical perspective.
About the Exercises
• Graded Exercise Sets Each exercise set begins with routine drill problems and progresses to problems
with more substance.
• True/False Exercises Most exercise sets end with a set of True/False exercises that are designed to check
conceptual understanding and logical reasoning. To avoid pure guessing, the students are required to justify
their responses in some way.
• Supplementary Exercise Sets Most chapters end with a set of supplementary exercises that tend to be
more challenging and force the student to draw on ideas from the entire chapter rather than a specific
section.
Howard Anton obtained his B.A. from Lehigh University, his M.A. from the University of Illinois, and his
Ph.D. from the Polytechnic University of Brooklyn, all in mathematics. In the early 1960s he worked for
Burroughs Corporation and Avco Corporation at Cape Canaveral, Florida, where he was involved with the
manned space program. In 1968 he joined the Mathematics Department at Drexel University, where he taught
full time until 1983. Since then he has devoted the majority of his time to textbook writing and activities for
mathematical associations. Dr. Anton was president of the EPADEL Section of the Mathematical Association
of America (MAA), served on the Board of Governors of that organization, and guided the creation of the
Student Chapters of the MAA. In addition to various pedagogical articles, he has published numerous
research papers in functional analysis, approximation theory, and topology. He is best known for his textbooks
in mathematics, which are among the most widely used in the world. There are currently more than 150
versions of his books, including translations into Spanish, Arabic, Portuguese, Italian, Indonesian, French,
Japanese, Chinese, Hebrew, and German. For relaxation, Dr. Anton enjoys travel and photography.
Copyright © 2010 John Wiley & Sons, Inc. All rights reserved.
, Preface
This edition of Elementary Linear Algebra gives an introductory treatment of linear algebra that is suitable for
a first undergraduate course. Its aim is to present the fundamentals of linear algebra in the clearest possible
way—sound pedagogy is the main consideration. Although calculus is not a prerequisite, there is some
optional material that is clearly marked for students with a calculus background. If desired, that material can
be omitted without loss of continuity.
Technology is not required to use this text, but for instructors who would like to use MATLAB, Mathematica,
Maple, or calculators with linear algebra capabilities, we have posted some supporting material that can be
accessed at either of the following Web sites:
www.howardanton.com
www.wiley.com/college/anton
Summary of Changes in this Edition
This edition is a major revision of its predecessor. In addition to including some new material, some of the old
material has been streamlined to ensure that the major topics can all be covered in a standard course. These
are the most significant changes:
• Vectors in 2-space, 3-space, and n-space Chapters 3 and 4 of the previous edition have been combined
into a single chapter. This has enabled us to eliminate some duplicate exposition and to juxtapose concepts
in n-space with those in 2-space and 3-space, thereby conveying more clearly how n-space ideas generalize
those already familiar to the student.
• New Pedagogical Elements Each section now ends with a Concept Review and a Skills mastery that
provide the student a convenient reference to the main ideas in that section.
• New Exercises Many new exercises have been added, including a set of True/False exercises at the end of
most sections.
• Earlier Coverage of Eigenvalues and Eigenvectors The chapter on eigenvalues and eigenvectors, which
was Chapter 7 in the previous edition, is Chapter 5 in this edition.
• Complex Vector Spaces The chapter entitled Complex Vector Spaces in the previous edition has been
completely revised. The most important ideas are now covered in Section 5.3 and Section 7.5 in the context
of matrix diagonalization. A brief review of complex numbers is included in the Appendix.
• Quadratic Forms This material has been extensively rewritten to focus more precisely on the most
important ideas.
• New Chapter on Numerical Methods In the previous edition an assortment of topics appeared in the last
chapter. That chapter has been replaced by a new chapter that focuses exclusively on numerical methods of
linear algebra. We achieved this by moving those topics not concerned with numerical methods elsewhere
in the text.
• Singular-Value Decomposition In recognition of its growing importance, a new section on Singular-Value
Decomposition has been added to the chapter on numerical methods.
, • Internet Search and the Power Method A new section on the Power Method and its application to
Internet search engines has been added to the chapter on numerical methods.
• Applications There is an expanded version of this text by Howard Anton and Chris Rorres entitled
Elementary Linear Algebra: Applications Version, 10th (ISBN 9780470432051), whose purpose is to
supplement this version with an extensive body of applications. However, to accommodate instructors who
asked us to include some applications in this version of the text, we have done so. These are generally less
detailed than those appearing in the Anton/Rorres text and can be omitted without loss of continuity.
Hallmark Features
• Relationships Among Concepts One of our main pedagogical goals is to convey to the student that linear
algebra is a cohesive subject and not simply a collection of isolated definitions and techniques. One way in
which we do this is by using a crescendo of Equivalent Statements theorems that continually revisit
relationships among systems of equations, matrices, determinants, vectors, linear transformations, and
eigenvalues. To get a general sense of how we use this technique see Theorems 1.5.3, 1.6.4, 2.3.8, 4.8.10,
4.10.4 and then Theorem 5.1.6, for example.
• Smooth Transition to Abstraction Because the transition from Rn to general vector spaces is difficult for
many students, considerable effort is devoted to explaining the purpose of abstraction and helping the
student to “visualize” abstract ideas by drawing analogies to familiar geometric ideas.
• Mathematical Precision When reasonable, we try to be mathematically precise. In keeping with the level
of student audience, proofs are presented in a patient style that is tailored for beginners. There is a brief
section in the Appendix on how to read proof statements, and there are various exercises in which students
are guided through the steps of a proof and asked for justification.
• Suitability for a Diverse Audience This text is designed to serve the needs of students in engineering,
computer science, biology, physics, business, and economics as well as those majoring in mathematics.
• Historical Notes To give the students a sense of mathematical history and to convey that real people
created the mathematical theorems and equations they are studying, we have included numerous Historical
Notes that put the topic being studied in historical perspective.
About the Exercises
• Graded Exercise Sets Each exercise set begins with routine drill problems and progresses to problems
with more substance.
• True/False Exercises Most exercise sets end with a set of True/False exercises that are designed to check
conceptual understanding and logical reasoning. To avoid pure guessing, the students are required to justify
their responses in some way.
• Supplementary Exercise Sets Most chapters end with a set of supplementary exercises that tend to be
more challenging and force the student to draw on ideas from the entire chapter rather than a specific
section.
