# A First Course in Linear Algebra

Ken Kuttler, Brigham Young University

Pub Date: 2017

Publisher: Lyryx

Language: English

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## Conditions of Use

Attribution

CC BY

## Reviews

This book covers a very large and comprehensive list of topics. Aside from the leading topics in a standard linear algebra course, there are some less-standard but highly important topics covered, such as spectral theory, abstract vector spaces,... read more

For the book's stated purpose of providing a first approach to linear algebra is met. The rigor is appropriate and the author has gone to great lengths to cover the standard definitions, theorems, and examples that are at the heart of linear... read more

The book is sufficiently comprehensive for its purpose as a first course in the subject. The row reduced echelon form and its many consequences and applications are covered well in the first several chapters. Later chapters on linear... read more

This book contains all of the material that would generally be covered in a Freshman or Sophomore Linear Algebra course. The section on vectors is quite extensive, and would be excellent to use in a Freshman course that needed to introduce vectors... read more

In my experience, text book works extremely well with the learning outcomes defined by my institution for entry level linear algebra course. For my students, textbook provides a foundation for the course. Techniques to solve the problems are easy... read more

The book includes all the topics we require in our introductory linear algebra course. read more

This text covers all the material an instructor could want to include in an introductory Linear Algebra course. The first three chapters (Systems of Equations, Matrices, and Determinants) are standard in any introductory Linear Algebra course,... read more

## Table of Contents

1 Systems of Equations

- 1.1 Systems of Equations, Geometry
- 1.2 Systems Of Equations, Algebraic Procedures

2 Matrices

- 2.1 Matrix Arithmetic
- 2.2 LU Factorization

3 Determinants

- 3.1 Basic Techniques and Properties
- 3.2 Applications of the Determinant

4 R^n

- 4.1 Vectors in R^n
- 4.2 Algebra in R^n
- 4.3 Geometric Meaning of Vector Addition
- 4.4 Length of a Vector
- 4.5 Geometric Meaning of Scalar Multiplication
- 4.6 Parametric Lines
- 4.7 The Dot Product
- 4.8 Planes in R^n
- 4.9 The Cross Product
- 4.10 Spanning, Linear Independence and Basis in R^n
- 4.11 Orthogonality and the Gram Schmidt Process
- 4.12 Applications

5 Linear Transformations

- 5.1 Linear Transformations
- 5.2 The Matrix of a Linear Transformation I
- 5.3 Properties of Linear Transformations
- 5.4 Special Linear Transformations in R^2
- 5.5 One to One and Onto Transformations
- 5.6 Isomorphisms
- 5.7 The Kernel And Image Of A Linear Map
- 5.8 The Matrix of a Linear Transformation II
- 5.9 The General Solution of a Linear System

6 Complex Numbers

- 6.1 Complex Numbers
- 6.2 Polar Form
- 6.3 Roots of Complex Numbers
- 6.4 The Quadratic Formula

7 Spectral Theory

- 7.1 Eigenvalues and Eigenvectors of a Matrix
- 7.2 Diagonalization
- 7.3 Applications of Spectral Theory
- 7.4 Orthogonality

8 Some Curvilinear Coordinate Systems

- 8.1 Polar Coordinates and Polar Graphs
- 8.2 Spherical and Cylindrical Coordinates

9 Vector Spaces

- 9.1 Algebraic Considerations
- 9.2 Spanning Sets
- 9.3 Linear Independence
- 9.4 Subspaces and Basis
- 9.5 Sums and Intersections
- 9.6 Linear Transformations
- 9.7 Isomorphisms
- 9.8 The Kernel And Image Of A Linear Map
- 9.9 The Matrix of a Linear Transformation

A Some Prerequisite Topics

- A.1 Sets and Set Notation
- A.2 Well Ordering and Induction

B Selected Exercise Answers

## About the Book

This text, originally by K. Kuttler, has been redesigned by the Lyryx editorial team as a first course in linear algebra for science and engineering students who have an understanding of basic algebra.

All major topics of linear algebra are available in detail, as well as proofs of important theorems. In addition, connections to topics covered in advanced courses are introduced. The text is designed in a modular fashion to maximize flexibility and facilitate adaptation to a given course outline and student profile.

Each chapter begins with a list of student learning outcomes, and examples and diagrams are given throughout the text to reinforce ideas and provide guidance on how to approach various problems. Suggested exercises are included at the end of each section, with selected answers at the end of the text.

Lyryx develops and supports open texts, with editorial services to adapt the text for each particular course. In addition, Lyryx provides content-specific formative online assessment, a wide variety of supplements, and in-house support available 7 days/week for both students and instructors.

## About the Contributors

### Author

**Ken Kuttler, **Professor of Mathematics at Bringham Young University. University of Texas at Austin, Ph.D. in Mathematics.