Softcover ISBN: | 978-1-4704-6295-6 |
Product Code: | TEXT/66 |
List Price: | $89.00 |
MAA Member Price: | $66.75 |
AMS Member Price: | $66.75 |
eBook ISBN: | 978-1-4704-6493-6 |
Product Code: | TEXT/66.E |
List Price: | $89.00 |
MAA Member Price: | $66.75 |
AMS Member Price: | $66.75 |
Softcover ISBN: | 978-1-4704-6295-6 |
eBook: ISBN: | 978-1-4704-6493-6 |
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 |
Softcover ISBN: | 978-1-4704-6295-6 |
Product Code: | TEXT/66 |
List Price: | $89.00 |
MAA Member Price: | $66.75 |
AMS Member Price: | $66.75 |
eBook ISBN: | 978-1-4704-6493-6 |
Product Code: | TEXT/66.E |
List Price: | $89.00 |
MAA Member Price: | $66.75 |
AMS Member Price: | $66.75 |
Softcover ISBN: | 978-1-4704-6295-6 |
eBook ISBN: | 978-1-4704-6493-6 |
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 |
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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.
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Table of Contents
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Cover
-
Title page
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Copyright
-
Contents
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Preface
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Chapter 1. Vector Spaces
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1.1. Sets and Logic
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1.2. Basic Definitions
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1.3. Properties of Vector Spaces
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1.4. Subtraction and Cancellation
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1.5. Euclidean Spaces
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1.6. Matrices
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1.7. Function Spaces
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1.8. Subspaces
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1.9. Lines and Planes
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Project: Quotient Spaces
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Project: Vector Fields
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Summary: Chapter 1
-
Review Exercises: Chapter 1
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Chapter 2. Systems of Linear Equations
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2.1. Notation and Terminology
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2.2. Gaussian Elimination
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2.3. Solving Linear Systems
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2.4. Applications
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Project: Numerical Methods
-
Project: Further Applications of Linear Systems
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Summary: Chapter 2
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Review Exercises: Chapter 2
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Chapter 3. Dimension Theory
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3.1. Linear Combinations
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3.2. Span
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3.3. Linear Independence
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3.4. Basis
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3.5. Dimension
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3.6. Coordinates
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Project: Infinite-Dimensional Vector Spaces
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Project: Linear Codes
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Summary: Chapter 3
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Review Exercises: Chapter 3
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Chapter 4. Inner Product Spaces
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4.1. Inner Products and Norms
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4.2. Geometry in Euclidean Spaces
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4.3. The Cauchy-Schwarz Inequality
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4.4. Orthogonality
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4.5. Fourier Analysis
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Project: Continuity
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Project: Orthogonal Polynomials
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Summary: Chapter 4
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Review Exercises: Chapter 4
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Chapter 5. Matrices
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5.1. Matrix Algebra
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5.2. Inverses
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5.3. Markov Chains
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5.4. Absorbing Markov Chains
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Project: Series of Matrices
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Project: Linear Models
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Summary: Chapter 5
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Review Exercises: Chapter 5
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Chapter 6. Linearity
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6.1. Linear Functions
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6.2. Compositions and Inverses
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6.3. Matrix of a Linear Function
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6.4. Matrices of Compositions and Inverses
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6.5. Change of Basis
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6.6. Image and Kernel
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6.7. Rank and Nullity
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6.8. Isomorphism
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Project: Dual Spaces
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Project: Iterated Function Systems
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Summary: Chapter 6
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Review Exercises: Chapter 6
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Chapter 7. Determinants
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7.1. Mathematical Induction
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7.2. Definition
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7.3. Properties of Determinants
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7.4. Cramer’s Rule
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7.5. Cross Product
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7.6. Orientation
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Project: Alternative Definitions
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Project: Curve Fitting with Determinants
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Summary: Chapter 7
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Review Exercises: Chapter 7
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Chapter 8. Eigenvalues and Eigenvectors
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8.1. Definitions
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8.2. Similarity
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8.3. Diagonalization
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8.4. Symmetric Matrices
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8.5. Systems of Differential Equations
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Project: Graph Theory
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Project: Numerical Methods for Eigenvalues and Eigenvectors
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Summary: Chapter 8
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Review Exercises: Chapter 8
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Hints and Answers to Selected Exercises
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Index
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Back Cover
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Additional Material
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RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a courseAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- Additional Material
- Requests
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: Infinite-Dimensional 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 Cauchy-Schwarz 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