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Hardcover ISBN:  9781470450847 
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Hardcover ISBN:  9781470450847 
Product Code:  AMSTEXT/42 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
eBook ISBN:  9781470453787 
Product Code:  AMSTEXT/42.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9781470450847 
eBook ISBN:  9781470453787 
Product Code:  AMSTEXT/42.B 
List Price:  $174.00 $131.50 
MAA Member Price:  $156.60 $118.35 
AMS Member Price:  $139.20 $105.20 

Book DetailsPure and Applied Undergraduate TextsVolume: 42; 2019; 389 ppMSC: Primary 15
Linear Algebra for the Young Mathematician is a careful, thorough, and rigorous introduction to linear algebra. It adopts a conceptual point of view, focusing on the notions of vector spaces and linear transformations, and it takes pains to provide proofs that bring out the essential ideas of the subject. It begins at the beginning, assuming no prior knowledge of the subject, but goes quite far, and it includes many topics not usually treated in introductory linear algebra texts, such as Jordan canonical form and the spectral theorem. While it concentrates on the finitedimensional case, it treats the infinitedimensional case as well. The book illustrates the centrality of linear algebra by providing numerous examples of its application within mathematics. It contains a wide variety of both conceptual and computational exercises at all levels, from the relatively straightforward to the quite challenging.
Readers of this book will not only come away with the knowledge that the results of linear algebra are true, but also with a deep understanding of why they are true.
ReadershipUndergraduate students interested in learning and teaching linear algebra.

Table of Contents

Cover

Title page

Preface

Part I . Vector spaces

Chapter 1. The basics

1.1. The vector space Fⁿ

1.2. Linear combinations

1.3. Matrices and the equation 𝐴𝑥=𝑏

1.4. The basic counting theorem

1.5. Matrices and linear transformations

1.6. Exercises

Chapter 2. Systems of linear equations

2.1. The geometry of linear systems

2.2. Solving systems of equations—setting up

2.3. Solving linear systems—echelon forms

2.4. Solving systems of equations—the reduction process

2.5. Drawing some consequences

2.6. Exercises

Chapter 3. Vector spaces

3.1. The notion of a vector space

3.2. Linear combinations

3.3. Bases and dimension

3.4. Subspaces

3.5. Affine subspaces and quotient vector spaces

3.6. Exercises

Chapter 4. Linear transformations

4.1. Linear transformations I

4.2. Matrix algebra

4.3. Linear transformations II

4.4. Matrix inversion

4.5. Looking back at calculus

4.6. Exercises

Chapter 5. More on vector spaces and linear transformations

5.1. Subspaces and linear transformations

5.2. Dimension counting and applications

5.3. Bases and coordinates: vectors

5.4. Bases and matrices: linear transformations

5.5. The dual of a vector space

5.6. The dual of a linear transformation

5.7. Exercises

Chapter 6. The determinant

6.1. Volume functions

6.2. Existence, uniqueness, and properties of the determinant

6.3. A formula for the determinant

6.4. Practical evaluation of determinants

6.5. The classical adjoint and Cramer’s rule

6.6. Jacobians

6.7. Exercises

Chapter 7. The structure of a linear transformation

7.1. Eigenvalues, eigenvectors, and generalized eigenvectors

7.2. Polynomials in cT

7.3. Application to differential equations

7.4. Diagonalizable linear transformations

7.5. Structural results

7.6. Exercises

Chapter 8. Jordan canonical form

8.1. Chains, Jordan blocks, and the (labelled) eigenstructure picture of cT

8.2. Proof that cT has a Jordan canonical form

8.3. An algorithm for Jordan canonical form and a Jordan basis

8.4. Application to systems of firstorder differential equations

8.5. Further results

8.6. Exercises

Part II . Vector spaces with additional structure

Chapter 9. Forms on vector spaces

9.1. Forms in general

9.2. Usual types of forms

9.3. Classifying forms I

9.4. Classifying forms II

9.5. The adjoint of a linear transformation

9.6. Applications to algebra and calculus

9.7. Exercises

Chapter 10. Inner product spaces

10.1. Definition, examples, and basic properties

10.2. Subspaces, complements, and bases

10.3. Two applications: symmetric and Hermitian forms, and the singular value decomposition

10.4. Adjoints, normal linear transformations, and the spectral theorem

10.5. Exercises

Appendix A. Fields

A.1. The notion of a field

A.2. Fields as vector spaces

Appendix B. Polynomials

B.1. Statement of results

B.2. Proof of results

Appendix C. Normed vector spaces and questions of analysis

C.1. Spaces of sequences

C.2. Spaces of functions

Appendix D. A guide to further reading

Index

Back Cover


Additional Material

Reviews

I enjoyed this book. It contains as clear an exposition of the JCF as I've seen anywhere, many applications which illustrate how transparent certain facts from calculus and differential equations are when viewed in the context of linear algebra and is very wellwritten.
Benjamin Linowitz, Oberlin College, MAA Reviews


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 Book Details
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Linear Algebra for the Young Mathematician is a careful, thorough, and rigorous introduction to linear algebra. It adopts a conceptual point of view, focusing on the notions of vector spaces and linear transformations, and it takes pains to provide proofs that bring out the essential ideas of the subject. It begins at the beginning, assuming no prior knowledge of the subject, but goes quite far, and it includes many topics not usually treated in introductory linear algebra texts, such as Jordan canonical form and the spectral theorem. While it concentrates on the finitedimensional case, it treats the infinitedimensional case as well. The book illustrates the centrality of linear algebra by providing numerous examples of its application within mathematics. It contains a wide variety of both conceptual and computational exercises at all levels, from the relatively straightforward to the quite challenging.
Readers of this book will not only come away with the knowledge that the results of linear algebra are true, but also with a deep understanding of why they are true.
Undergraduate students interested in learning and teaching linear algebra.

Cover

Title page

Preface

Part I . Vector spaces

Chapter 1. The basics

1.1. The vector space Fⁿ

1.2. Linear combinations

1.3. Matrices and the equation 𝐴𝑥=𝑏

1.4. The basic counting theorem

1.5. Matrices and linear transformations

1.6. Exercises

Chapter 2. Systems of linear equations

2.1. The geometry of linear systems

2.2. Solving systems of equations—setting up

2.3. Solving linear systems—echelon forms

2.4. Solving systems of equations—the reduction process

2.5. Drawing some consequences

2.6. Exercises

Chapter 3. Vector spaces

3.1. The notion of a vector space

3.2. Linear combinations

3.3. Bases and dimension

3.4. Subspaces

3.5. Affine subspaces and quotient vector spaces

3.6. Exercises

Chapter 4. Linear transformations

4.1. Linear transformations I

4.2. Matrix algebra

4.3. Linear transformations II

4.4. Matrix inversion

4.5. Looking back at calculus

4.6. Exercises

Chapter 5. More on vector spaces and linear transformations

5.1. Subspaces and linear transformations

5.2. Dimension counting and applications

5.3. Bases and coordinates: vectors

5.4. Bases and matrices: linear transformations

5.5. The dual of a vector space

5.6. The dual of a linear transformation

5.7. Exercises

Chapter 6. The determinant

6.1. Volume functions

6.2. Existence, uniqueness, and properties of the determinant

6.3. A formula for the determinant

6.4. Practical evaluation of determinants

6.5. The classical adjoint and Cramer’s rule

6.6. Jacobians

6.7. Exercises

Chapter 7. The structure of a linear transformation

7.1. Eigenvalues, eigenvectors, and generalized eigenvectors

7.2. Polynomials in cT

7.3. Application to differential equations

7.4. Diagonalizable linear transformations

7.5. Structural results

7.6. Exercises

Chapter 8. Jordan canonical form

8.1. Chains, Jordan blocks, and the (labelled) eigenstructure picture of cT

8.2. Proof that cT has a Jordan canonical form

8.3. An algorithm for Jordan canonical form and a Jordan basis

8.4. Application to systems of firstorder differential equations

8.5. Further results

8.6. Exercises

Part II . Vector spaces with additional structure

Chapter 9. Forms on vector spaces

9.1. Forms in general

9.2. Usual types of forms

9.3. Classifying forms I

9.4. Classifying forms II

9.5. The adjoint of a linear transformation

9.6. Applications to algebra and calculus

9.7. Exercises

Chapter 10. Inner product spaces

10.1. Definition, examples, and basic properties

10.2. Subspaces, complements, and bases

10.3. Two applications: symmetric and Hermitian forms, and the singular value decomposition

10.4. Adjoints, normal linear transformations, and the spectral theorem

10.5. Exercises

Appendix A. Fields

A.1. The notion of a field

A.2. Fields as vector spaces

Appendix B. Polynomials

B.1. Statement of results

B.2. Proof of results

Appendix C. Normed vector spaces and questions of analysis

C.1. Spaces of sequences

C.2. Spaces of functions

Appendix D. A guide to further reading

Index

Back Cover

I enjoyed this book. It contains as clear an exposition of the JCF as I've seen anywhere, many applications which illustrate how transparent certain facts from calculus and differential equations are when viewed in the context of linear algebra and is very wellwritten.
Benjamin Linowitz, Oberlin College, MAA Reviews