Preface

This linear algebra textbook was originally designed to be presented as twenty five, fifty minute lectures suitable for sophomores likely to use the material for applications but still requiring a solid foundation in this fundamental branch of mathematics. The main idea of the course is to emphasize the concepts of vector spaces and linear transformations as mathematical structures that can be used to model the world around us. Once ``persuaded'' of this truth, students learn explicit skills such as Gaussian elimination and diagonalization in order that vectors and linear transformations become calculational tools, rather than abstract mathematics.

In practical terms, the course aims to produce students who can perform computations with large linear systems while at the same time understand the concepts behind these techniques. Often-times when a problem can be reduced to one of linear algebra it is ``solved''. These notes do not devote much space to applications (there are already a plethora of textbooks with titles involving some permutation of the words ``linear'', ``algebra'' and ``applications''). Instead, they attempt to explain the fundamental concepts carefully enough that students will realize for their own selves when the particular application they encounter in future studies is ripe for a solution via linear algebra.

The notes are designed to be used in conjunction with a set of online homework exercises which help the students read the lecture notes and learn basic linear algebra skills. Interspersed among the lecture notes are links to simple online problems that test whether students are actively reading the notes. In addition there are two sets of sample midterm problems with solutions as well as a sample final exam. There are also a set of ten online assignments which are collected weekly. The first assignment is designed to ensure familiarity with some basic mathematic notions (sets, functions, logical quantifiers and basic methods of proof). The remaining nine assignments are devoted to the usual matrix and vector gymnastics expected from any sophomore linear algebra class. These exercises are all available here.
Webwork is an open source, online homework system which originated at the University of Rochester. It can efficiently check whether a student has answered an explicit, typically computation-based, problem correctly. The problem sets chosen to accompany these notes could contribute roughly 20% of a student's grade, and ensure that basic computational skills are mastered. Most students rapidly realize that it is best to print out the Webwork assignments and solve them on paper before entering the answers online. Those who do not tend to fare poorly on midterm examinations. We have found that there tend to be relatively few questions from students in office hours about the Webwork assignments. Instead, by assigning 20% of the grade to written assignments drawn from problems chosen randomly from the review exercises at the end of each lecture, the student's focus was primarily on understanding ideas. They range from simple tests of understanding of the material in the lectures to more difficult problems, all of them require thinking, rather than blind application of mathematical "recipes". Office hour questions reflected this and offered an excellent chance to give students tips how to present written answers in a way that would convince the person grading their work that they deserved full credit!

Each lecture concludes with references to the comprehensive online textbooks of Jim Hefferon and Rob Beezer:

• http://joshua.smcvt.edu/linearalgebra/
• http://linear.ups.edu/index.html

• The Khan Academy
• MIT OpenCourseWare, Professor Gilbert Strang

• "Introductory Linear Algebra, An Applied First Course", B. Kolman and D. Hill, Pearson 2001.
• "Linear Algebra and Its Applications", David C. Lay, Addison--Weseley 2011.
• "Introduction to Linear Algebra", Gilbert Strang, Wellesley Cambridge Press 2009.
• "Linear Algebra Done Right", S. Axler, Springer 1997.
• "Algebra and Geometry", D. Holten and J. Lloyz, CBRC, 1978.
• "Schaum's Outline of Linear Algebra", S. Lipschutz and M. Lipson, McGraw-Hill 2008.

There are still many errors in the notes, as well as awkwardly explained concepts. An army of 400 students, Fu Liu, Stephen Pon and Gerry Puckett have already found many of them. Rohit Thomas has spent a great deal of time editing these notes and has improved them immeasurably. We also thank Captain Conundrum for providing us his solutions to the sample midterm and final questions. The review exercises would provide a better survey of what linear algebra really is if there were more ``applied'' questions. We welcome your contributions!