# A First Course on Linear Algebra

Written by: Robert A Beezer (Puget Sound)

The author of this free online linear algebra textbook, Robert A. Beezer, has taught linear algebra 39 times (as of May 2014) and has been teaching undergraduate mathematics courses since 1978. He is currently a member of the faculty at the University of Puget Sound.

Beezer lists courses in differential and integral calculus and perhaps some multivariate calculus as prerequisites but reassures students that calculus knowledge is not truly required to be successful with his text. He explains that students will need a “level of mathematical maturity” gained from those courses moreso than actual knowledge of calculus.

Most of the exercises have complete solutions. Those which do not are identified clearly within the text. Beezer encourages students to not peek at the answers too soon and instead try to work the exercises diligently to better master the material.

The text is offered as an online text and a downloadable PDF. However, according to Beezer, the online version is the most complete.

Systems of Linear Equations
What is Linear Algebra?
Solving Systems of Linear Equations
Reduced Row-Echelon Form
Types of Solution Sets
Homogeneous Systems of Equations
Nonsingular Matrices
Vectors
Vector Operations
Linear Combinations
Spanning Sets
Linear Independence
Linear Dependence and Spans
Orthogonality
Matrices
Matrix Operations
Matrix Multiplication
Matrix Inverses and Systems of Linear Equations
Matrix Inverses and Nonsingular Matrices
Column and Row Spaces
Four Subsets
Determinants
Determinant of a Matrix
Properties of Determinants of Matrices
Eigenvalues
Eigenvalues and Eigenvectors
Properties of Eigenvalues and Eigenvectors
Similarity and Diagonalization
Linear Transformations
Linear Transformations
Injective Linear Transformations
Surjective Linear Transformations
Invertible Linear Transformations
Representations
Vector Representations
Matrix Representations
Change of Basis
Orthonormal Diagonalization
Preliminaries
Complex Number Operations
Sets
Archetypes
A-Z
Reference
Notation
Definitions
Theorems
Diagrams
Examples
Sage
Proof Techniques 