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.
Author(s): Robert Messer
Edition: 2
Publisher: American Mathematical Society
Year: 2021
Language: English
Pages: 420
Tags: mathematics, algebra, linear, 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
