# A First Course in Linear Algebra

This open textbook, originally by K. Kuttler, has been redesigned by the Lyryx editorial team as a first course for the general students who have an understanding of basic high school algebra and intend to be users of linear algebra methods in their profession, from business & economics to science students. All major topics of linear algebra are available in detail, as well as justifications of important results. In addition, connections to topics covered in advanced courses are introduced. The textbook 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 textbook.
License: Creative Commons Attribution. This license is considered to be some to be the most open license since it is the least restrictive. It allows reuse, remixing, and distribution (including commercial), only requiring attribution. The content can be remixed into content of other license, but on the other hand it allows the remix to be put under a more restrictive license.
Formats:
• PDF. A Portable Document Format (PDF) file is can be opened using the free Acrobat Reader. It is not an editable format.
• TeX. A TeX file use the TeX or LaTeX typesetting engine. TeX software is available free for most platforms. It is an editable format
Openness Rating (0-4): 2
Openness Comments: The TeX source may be available from Lyryx Learning.
1. Systems of Equations
1. Systems of Equations, Geometry
2. Systems Of Equations, Algebraic Procedures
2. Matrices
1. Matrix Arithmetic
2. LU Factorization
3. Determinants
1. Basic Techniques and Properties
2. Applications of the Determinant
4. R^n
1. Vectors in R^n
2. Algebra in R^n
3. Geometric Meaning of Vector Addition
4. Length of a Vector
5. Geometric Meaning of Scalar Multiplication
6. Parametric Lines
7. The Dot Product
8. Planes in R^n
9. The Cross Product
10. Spanning, Linear Independence and Basis in R^n
11. Orthogonality and the Gram Schmidt Process
12. Applications
5. Linear Transformations
1. Linear Transformations
2. The Matrix of a Linear Transformation I
3. Properties of Linear Transformations
4. Special Linear Transformations in R^2
5. One to One and Onto Transformations
6. Isomorphisms
7. The Kernel And Image Of A Linear Map
8. The Matrix of a Linear Transformation II
9. The General Solution of a Linear System
6. Complex Numbers
1. Complex Numbers
2. Polar Form
3. Roots of Complex Numbers
7. Spectral Theory
1. Eigenvalues and Eigenvectors of a Matrix
2. Diagonalization
3. Applications of Spectral Theory
4. Orthogonality
8. Some Curvilinear Coordinate Systems
1. Polar Coordinates and Polar Graphs
2. Spherical and Cylindrical Coordinates
9. Vector Spaces
1. Algebraic Considerations
2. Spanning Sets
3. Linear Independence
4. Subspaces and Basis
5. Sums and Intersections
6. Linear Transformations
7. Isomorphisms
8. The Kernel And Image Of A Linear Map
9. The Matrix of a Linear Transformation
Supplements:
• MyOpenMath / Lumen OHM online homework. MyOpenMath is a free online homework system, built on the open source IMathAS assessment platform. It provides randomized, algorithmically generated homework with automated grading of numerical and algebraic answers, similar to WebAssign and MyMathLab. It also provides a course management system with gradebook, file posting, discussion forums, etc. Assessment sets have been created for this textbook, which may be available for self-study by students, or can copied as a starter course shell by faculty.

MyOpenMath use is free with community support through forums. For Washington State faculty, the WAMAP.org site also mirrors this content.

Lumen OHM is a commercial alternative to MyOpenMath that provides support for faculty and large scale adoption and administration, service level agreements, and additional curated course bundles.
Notes: Slides and online homework are also available from Lyryx Learning.