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Softcover ISBN:  9781470462956 
Product Code:  TEXT/66 
List Price:  $89.00 
MAA Member Price:  $66.75 
AMS Member Price:  $66.75 
Sale Price:  $57.85 
eBook ISBN:  9781470464936 
Product Code:  TEXT/66.E 
List Price:  $89.00 
MAA Member Price:  $66.75 
AMS Member Price:  $66.75 
Sale Price:  $57.85 
Softcover ISBN:  9781470462956 
eBook ISBN:  9781470464936 
Product Code:  TEXT/66.B 
List Price:  $178.00 $133.50 
MAA Member Price:  $133.50 $100.13 
AMS Member Price:  $133.50 $100.13 
Sale Price:  $115.70 $86.78 

Book DetailsAMS/MAA TextbooksVolume: 66; 2021; 420 ppMSC: Primary 15
Linear Algebra: Gateway to Mathematics uses linear algebra as a vehicle to introduce students to the inner workings of mathematics. The structures and techniques of mathematics in turn provide an accessible framework to illustrate the powerful and beautiful results about vector spaces and linear transformations.
The unifying concepts of linear algebra reveal the analogies among three primary examples: Euclidean spaces, function spaces, and collections of matrices. Students are gently introduced to abstractions of higher mathematics through discussions of the logical structure of proofs, the need to translate terminology into notation, and efficient ways to discover and present proofs. Application of linear algebra and concrete examples tie the abstract concepts to familiar objects from algebra, geometry, calculus, and everyday life.
Students will finish a course using this text with an understanding of the basic results of linear algebra and an appreciation of the beauty and utility of mathematics. They will also be fortified with a degree of mathematical maturity required for subsequent courses in abstract algebra, real analysis, and elementary topology. Students who have prior background in dealing with the mechanical operations of vectors and matrices will benefit from seeing this material placed in a more general context.
ReadershipUndergraduate students interested in learning linear algebra.

Table of Contents

Cover

Title page

Copyright

Contents

Preface

Chapter 1. Vector Spaces

1.1. Sets and Logic

1.2. Basic Definitions

1.3. Properties of Vector Spaces

1.4. Subtraction and Cancellation

1.5. Euclidean Spaces

1.6. Matrices

1.7. Function Spaces

1.8. Subspaces

1.9. Lines and Planes

Project: Quotient Spaces

Project: Vector Fields

Summary: Chapter 1

Review Exercises: Chapter 1

Chapter 2. Systems of Linear Equations

2.1. Notation and Terminology

2.2. Gaussian Elimination

2.3. Solving Linear Systems

2.4. Applications

Project: Numerical Methods

Project: Further Applications of Linear Systems

Summary: Chapter 2

Review Exercises: Chapter 2

Chapter 3. Dimension Theory

3.1. Linear Combinations

3.2. Span

3.3. Linear Independence

3.4. Basis

3.5. Dimension

3.6. Coordinates

Project: InfiniteDimensional Vector Spaces

Project: Linear Codes

Summary: Chapter 3

Review Exercises: Chapter 3

Chapter 4. Inner Product Spaces

4.1. Inner Products and Norms

4.2. Geometry in Euclidean Spaces

4.3. The CauchySchwarz Inequality

4.4. Orthogonality

4.5. Fourier Analysis

Project: Continuity

Project: Orthogonal Polynomials

Summary: Chapter 4

Review Exercises: Chapter 4

Chapter 5. Matrices

5.1. Matrix Algebra

5.2. Inverses

5.3. Markov Chains

5.4. Absorbing Markov Chains

Project: Series of Matrices

Project: Linear Models

Summary: Chapter 5

Review Exercises: Chapter 5

Chapter 6. Linearity

6.1. Linear Functions

6.2. Compositions and Inverses

6.3. Matrix of a Linear Function

6.4. Matrices of Compositions and Inverses

6.5. Change of Basis

6.6. Image and Kernel

6.7. Rank and Nullity

6.8. Isomorphism

Project: Dual Spaces

Project: Iterated Function Systems

Summary: Chapter 6

Review Exercises: Chapter 6

Chapter 7. Determinants

7.1. Mathematical Induction

7.2. Definition

7.3. Properties of Determinants

7.4. Cramer’s Rule

7.5. Cross Product

7.6. Orientation

Project: Alternative Definitions

Project: Curve Fitting with Determinants

Summary: Chapter 7

Review Exercises: Chapter 7

Chapter 8. Eigenvalues and Eigenvectors

8.1. Definitions

8.2. Similarity

8.3. Diagonalization

8.4. Symmetric Matrices

8.5. Systems of Differential Equations

Project: Graph Theory

Project: Numerical Methods for Eigenvalues and Eigenvectors

Summary: Chapter 8

Review Exercises: Chapter 8

Hints and Answers to Selected Exercises

Index

Back Cover


Additional Material

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 Table of Contents
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Linear Algebra: Gateway to Mathematics uses linear algebra as a vehicle to introduce students to the inner workings of mathematics. The structures and techniques of mathematics in turn provide an accessible framework to illustrate the powerful and beautiful results about vector spaces and linear transformations.
The unifying concepts of linear algebra reveal the analogies among three primary examples: Euclidean spaces, function spaces, and collections of matrices. Students are gently introduced to abstractions of higher mathematics through discussions of the logical structure of proofs, the need to translate terminology into notation, and efficient ways to discover and present proofs. Application of linear algebra and concrete examples tie the abstract concepts to familiar objects from algebra, geometry, calculus, and everyday life.
Students will finish a course using this text with an understanding of the basic results of linear algebra and an appreciation of the beauty and utility of mathematics. They will also be fortified with a degree of mathematical maturity required for subsequent courses in abstract algebra, real analysis, and elementary topology. Students who have prior background in dealing with the mechanical operations of vectors and matrices will benefit from seeing this material placed in a more general context.
Undergraduate students interested in learning linear algebra.

Cover

Title page

Copyright

Contents

Preface

Chapter 1. Vector Spaces

1.1. Sets and Logic

1.2. Basic Definitions

1.3. Properties of Vector Spaces

1.4. Subtraction and Cancellation

1.5. Euclidean Spaces

1.6. Matrices

1.7. Function Spaces

1.8. Subspaces

1.9. Lines and Planes

Project: Quotient Spaces

Project: Vector Fields

Summary: Chapter 1

Review Exercises: Chapter 1

Chapter 2. Systems of Linear Equations

2.1. Notation and Terminology

2.2. Gaussian Elimination

2.3. Solving Linear Systems

2.4. Applications

Project: Numerical Methods

Project: Further Applications of Linear Systems

Summary: Chapter 2

Review Exercises: Chapter 2

Chapter 3. Dimension Theory

3.1. Linear Combinations

3.2. Span

3.3. Linear Independence

3.4. Basis

3.5. Dimension

3.6. Coordinates

Project: InfiniteDimensional Vector Spaces

Project: Linear Codes

Summary: Chapter 3

Review Exercises: Chapter 3

Chapter 4. Inner Product Spaces

4.1. Inner Products and Norms

4.2. Geometry in Euclidean Spaces

4.3. The CauchySchwarz Inequality

4.4. Orthogonality

4.5. Fourier Analysis

Project: Continuity

Project: Orthogonal Polynomials

Summary: Chapter 4

Review Exercises: Chapter 4

Chapter 5. Matrices

5.1. Matrix Algebra

5.2. Inverses

5.3. Markov Chains

5.4. Absorbing Markov Chains

Project: Series of Matrices

Project: Linear Models

Summary: Chapter 5

Review Exercises: Chapter 5

Chapter 6. Linearity

6.1. Linear Functions

6.2. Compositions and Inverses

6.3. Matrix of a Linear Function

6.4. Matrices of Compositions and Inverses

6.5. Change of Basis

6.6. Image and Kernel

6.7. Rank and Nullity

6.8. Isomorphism

Project: Dual Spaces

Project: Iterated Function Systems

Summary: Chapter 6

Review Exercises: Chapter 6

Chapter 7. Determinants

7.1. Mathematical Induction

7.2. Definition

7.3. Properties of Determinants

7.4. Cramer’s Rule

7.5. Cross Product

7.6. Orientation

Project: Alternative Definitions

Project: Curve Fitting with Determinants

Summary: Chapter 7

Review Exercises: Chapter 7

Chapter 8. Eigenvalues and Eigenvectors

8.1. Definitions

8.2. Similarity

8.3. Diagonalization

8.4. Symmetric Matrices

8.5. Systems of Differential Equations

Project: Graph Theory

Project: Numerical Methods for Eigenvalues and Eigenvectors

Summary: Chapter 8

Review Exercises: Chapter 8

Hints and Answers to Selected Exercises

Index

Back Cover