# 22 Mathematics, Calculus and Statistics

# A First Course in Linear Algebra

Description: A First Course in Linear Algebra is an introductory textbook designed for university sophomores and juniors. Typically such a student will have taken calculus, but this is not a prerequisite. The book begins with systems of linear equations, then covers matrix algebra, before taking up finite-dimensional vector spaces in full generality. The final chapter covers matrix representations of linear transformations, through diagonalization, change of basis and Jordan canonical form. Along the way, determinants and eigenvalues get fair time. There is a comprehensive online edition and PDF versions are available to download for printing or on-screen viewing.

This textbook has more freedom than most (but see some exceptions). First, there is no cost to acquire this text, and you are under no obligation whatsoever to compensate or donate to the author or publisher. So in this most basic sense, it is a free textbook. Therefore you can also make as many copies as you like, ensuring that the book will never go out-of-print. You may modify copies of the book for your own use – for example, you may wish to change to a prefered notation for certain objects or add a few new sections. I have applied a copyright to the book, and subsequently licensed it with a GNU Free Documentation License (GFDL). It is this combination that allows me to give you greater freedoms in how you use the text, thus liberating it from some of the antiquated notions of copyright that apply to books in physical form. The main caveat is that if you make modifications and then distribute a modified version, you are required to again apply the GFDL license to the result so that others may benefit from your modifications.

# A First Course in Linear Algebra: Lyryx

Description: A First Course in Linear Algebra, 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. This version was revised by the Lyryx editorial team in 2017.

**Includes**: Exercises with selected answers. Slides and question bank along with the source are available on request.

# Abstract Algebra: Theory and Applications

Description: Tom Judson’s Abstract Algebra: Theory and Applications is an open source textbook designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many nontrivial applications. The 2020 Annual Edition is now available. Electronic editions have been updated.

First, there is no cost to acquire this text, and you are under no obligation whatsoever to compensate or donate to the author or publisher. So in this most basic sense, it is a free textbook. Therefore you can also make as many copies as you like, ensuring that the book will never go out-of-print. You may modify copies of the book for your own use – for example, you may wish to change to a prefered notation for certain objects or add a few new sections. There is a copyright on the book, and subsequently it is licensed with a GNU Free Documentation License (GFDL). It is this combination that allows the author to give you greater freedoms in how you use the text, thus liberating it from some of the antiquated notions of copyright that apply to books in physical form. The main caveat is that if you make modifications and then distribute a modified version, you are required to again apply the GFDL license to the result so that others may benefit from your modifications.

# Active Calculus Single Variable

Description: This textbook was written by math professors from an American university.

**Includes**: interactive exercises and solutions, a lab manual, and other instructor resources.

# Active Calculus Multivariable

Description: This textbook is meant to complement the Single Variable edition of Active Calculus.

**Includes**: exercises and solutions.

# Algebra and Trignometry

# APEX Calculus

Description: This text, or portions of it, has been adopted at a considerable number of American institutions to teach calculus, according to their website. These institutions include the Virginia Military Institute, where the author is an associate professor of mathematics.

**Includes**: interactive graphics, exercises and solutions.

# Applied Discrete Structures

Description:** Applied Discrete Stuctures** by Al Doerr and Ken Levasseur is a *free* open content textbook. You can essentially share it with anyone as long as you leave the Creative Commons license in place. See a more precise legal description below. This textbook contains the content of a two semester course in discrete structures, which is typically a second-year course for students in computer science or mathematics, but it does not have a calculus prerequisite. The material for the first semester is in chapters 1-10 and includes logic, set theory, functions, relations, recursion, graphs, trees, and elementary combinatorics. The second semester material in chapters 11-16 deals with algebraic structures: binary operations, groups, matrix algebra, Boolean algebra, monoids and automata, rings and fields. The following are available for download:

- full pdf version Chapters 1-16, (8Mb)
- Part 1 – Fundamentals, Chapters 1-10, (5 Mb)
- Part 2 – Algebraic Structures, Chapters 11-16, (3.3 Mb)

The text is available in Version 3.7, May 2020. Source code for the current version is available at https://github.com/klevasseur/ads

One can view the textbook online at http://faculty.uml.edu/klevasseur/ads/index-ads.html

**Includes**: Exercises with short answers to odd problems, some concepts illustrated with Sage enabled web pages, blog page, Wiki, practice exams and videos.

# Basic Analysis: Introduction to Real Analysis

Description: This free online textbook (OER more formally) is a course in undergraduate real analysis (somewhere it is called “advanced calculus”). The book is meant both for a basic course for students who do not necessarily wish to go to graduate school, but also as a more advanced course that also covers topics such as metric spaces and should prepare students for graduate study. A prerequisite for the course is a basic proof course. An advanced course could be two semesters long with some of the second-semester topics such as multivariable differential calculus, path integrals, and the multivariable integral using the second volume. There are more topics than can be covered in two semesters, and it can also be reading for beginning graduate students to refresh their analysis or fill in some of the holes. This text was originally developed by Jiri Lebl in 2012 (Volume I). Both Volume I and II were updated in June 2020. The following are available for download:

- Volume I as pdf (Version 5.3, June 10th, 2020, 282 pages, 1.8 MB download)
- Volume II as pdf (Version 2.3, June 10th, 2020, 195 pages, 1.4 MB download)
- latex source code for both Volumes I & II from GitHub

**Includes**: Exercises with no solutions.

# Book of Proof

Description: This book is an introduction to the standard methods of proving mathematical theorems. It has been approved by the American Institute of Mathematics’ Open Textbook Initiative. Also see the Mathematical Association of America Math DL review (of the 1st edition) and the Amazon reviews. An adoptions list is here. This book was originally developed by Richard Hammack in 2018. Please note that the Creative Commons License for this text **does not permit**** altering** of content for anything other than personal use. A pdf version of the book is available for download from the book site.

**Includes**: Syllabus, calendar, sample tests and workbook.

# Calculus

Description: Originally published through traditional means, this textbook is now available as an open educational resource. This text includes single and multivariable calculus and was written by a professor of mathematics at the Massachusetts Institute of Technology.

**Includes**: an instructor’s manual, a student study guide with exercises and select solutions, and links to a calculus video series by the author.

# Calculus: Early Transcendentals

Description: Calculus: Early Transcendentals, originally by D. Guichard, has been redesigned by the Lyryx editorial team. Substantial portions of the content, examples, and diagrams have been redeveloped, with additional contributions provided by experienced and practicing instructors. This approachable textbook provides a comprehensive understanding of the necessary techniques and concepts of the typical Calculus course sequence, and is suitable for the standard Calculus I, II and III courses. The latex source can be requested from Lyryx Learning.

**Includes**: Exercises and answers, slides. Lyryx Learning also has some online homework available.

# Calculus for the Life Sciences: A Modelling Approach Volume 1

Description: This textbook was written by two professors at Iowa State University with the goal introducing students to scientific modeling. This course aims to teach students the necessary skills and concepts from a traditional physical sciences course, while explaining the applications of calculus to the life sciences.

**Includes**: exercises, solutions to select problems, and MATLAB exercises.

# Calculus: OpenStax Volumes 1-3

Volume 1 concepts: functions, limits, derivatives, and integration

Volume 2 concepts: integration, differential equations, sequences and series, and parametric equations and polar coordinates

Volume 3 concepts: parametric equations and polar coordinates, vectors, functions of several variables, multiple integration, and second-order differential equations

**Includes**: Exercises and answers. Volume 1 has been faculty-reviewed, adopted, is accessible, and has ancillary resources. Volume 2 has been adopted, has ancillary resources, and is accessible. Volume 3 is accessible and has ancillary resources. Adopt and adapt these volumes at the links above on Volumes 1, 2, and 3, respectively.

# Calculus Two

Description: Calculus Two is a Computer-Aided Problem-Solving Approach Using SageMath written by Charles Bergeron and other open source creators. This is a text for a second undergraduate Calculus course. This book includes applications from many disciplines, and integrates the use of the open-source computer algebra system SageMath throughout. This free online book should be usable as a stand-alone textbook or as a supplementary resource.

You can use, print, duplicate, share this book as much as you want. You can base your own book on it and reuse parts if you keep the license the same. If you plan to use it commercially (sell it for more than duplicating cost), then you need to contact me and we will work something out.

This work is based upon the open-source books *Community Calculus* by David Guichard et al. and *Mooculus* and *Mooculus: Sequences and Series* by Jim Fowler and Bart Snapp.

Major changes include:

- Creating a full chapter on linear algebra topics.
- Strengthening the presentation of topics that prepare students for a course in Differential Equations, including hyperbolic functions and complex numbers.
- Inserting instruction and exercises using the computer algebra system SageMath.
- Adding applications in healthcare, chemistry and pharmacy.

Further content was adapted from these open-source resources:

*Linear Algebra*by Jim Hefferon.*Precalculus*by Carl Stitz and Jeff Zeager.

I am grateful to all authors listed on the Attributions page, and indeed all open-source creators, for making their materials available. Without this publishing model, I probably would not have been able to to realize this project. What a nice way for us to *collaborate*! I am also grateful to the authors listed on the Further Reading page, for influencing my thinking on this subject, and its presentation.

The following are available for this resource:

- Download the textbook as a single PDF file, Spring 2017 edition (typeset on March 6, 2017)
- Progress packet (lists all learning activities for all class meeting times during the course)
- Syllabus for MAT 211

**Includes**: Exercises with solutions to selected exercises, example solutions using Sagemath computations, syllabus, progress packet, videos.

# Combinatorics

Description: This free undergraduate text book provides an introduction to enumeration, graph theory, and design theory. It is aimed at upper-level undergraduate students and the exercises expect some mathematical sophistication, including a reasonable ability to construct proofs. The text is designed to be used in an undergraduate course, but could be suitable for independent study by a student with some mathematical background and understanding of proofs. It does not assume any background knowledge of combinatorics. This text is written by Joy Morris of the University of Lethbridge.

The book is being released online with a Creative Commons license (Attribution-NonCommercial-ShareAlike 2.0). Although not in final form, it has already been used as a textbook for several semesters by 2 different instructors at the University of Lethbridge. The following are available:

- Download book in PDF format, June 2017 version (approximately 250 pages and 1.2 MB)
- Latex source available by contacting the author: joy.morris@uleth.ca

**Includes**: Exercises with solutions to selected exercises.

# Community Calculus (Single, Multivariable)

Description: This online compilation of four calculus textbooks is frequently updated and has been used in courses. It has been positively reviewed by the Mathematical Association of America. It also covers early and late transcendentals.

**Includes**: exercises, solutions, and WeBWorK problem sets.

# Contemporary Calculus

Description: This textbook was created as part of the Washington State Colleges’ Open Course Library Project, which received funding through the Gates Foundation. This text has been used by thousands of students within Washington State and was developed by a professor at Bellevue College.

**Includes**: exercises, solutions to odd-numbered problems, and other resources.

##### Development Versions, Source Code with Revision Control

Source code is available in a git repository, hosted at GitHub, and authored with PreTeXt. Prior to the 2015 edition, the source code was LaTeX and was primarily hosted at bitbucket.org.

Periodically a current snapshot of the book as a PDF is available here. These are not archived, nor tagged in the source code repository. They simply accumulate fixes or additions made to annual official releases. If there is nothing listed here, then use the current Annual Edition.

The absolute most recent version is available as source code through a git repository at github.com, the specific address is https://github.com/twjudson/aata. Links below to recent source code archives come from github.com, so the current version

links below are the most up-to-date version.

If you would like to participate in contributing to the book, you will want to use git. Then you can fork the repository on GitHub, or create a local copy of the source code by doing the following in a terminal window:

- Set your working directory to a place where a new directory named aata can live.
- Issue the command: git clone https://github.com/twjudson/aata.git

# Differential Calculus for the Life Sciences

Description: This textbook was written by a math professor at the University of British Columbia and is currently used as a textbook for introductory calculus courses for life sciences undergraduates.

**Includes**: exercises and solutions to selected problems.

# Differential Equations

Description: A first course on differential equations, aimed at engineering students. The prerequisite for the course is the basic calculus sequence. This free online book (OER more formally) should be usable as a stand-alone textbook or as a companion to a course using another book such as Edwards and Penney, *Differential Equations and Boundary Value Problems: Computing and Modeling* or Boyce and DiPrima, *Elementary Differential Equations and Boundary Value Problems* (section correspondence to these two is given). This book was originally developed by Jiri Lebl in 2012. The current version was updated July 21st, 2020. The following are available for download:

- complete book as a pdf file (July 21st, 2020, version 6.1, 466 pages, approximately 4.1 MB download)
- Browse the web version of the book (for easier reading on the web). This version uses PreTeXt and so should be easier to browse and read.
- all figures in one big zip file
- complete latex source code on GitHub
- WebWork a homework set containing currently 474 problems

**Includes**: Exercises (740) with some solutions (247), homework exercises using the WebWork system.

# Differential Equations

Description: This is a differential equations book including Linear Algebra Topics And Computer-Aided Problem-Solving Using Maxima or SageMath by Charles Bergeron, Jiri Lebl, and other open-source creators.

This is a text for a first undergraduate Differential Equations course. It does not assume previous coverage of Linear Algebra. This book includes applications from many disciplines, and integrates the use of the open-source computer algebra system Maxima or SageMath throughout. This free online book should be usable as a stand-alone textbook or as a supplementary resource.

This work is heavily based upon the open-source book *Notes on Diffy Qs: Differential Equations for Engineers* by Jiri Lebl. That book has been selected as an Approved Textbook in the American Institute of Mathematics Open Textbook Initiative.

Major changes include:

- Stripping the chapters on PDEs and nonlinear systems.
- Creating a full chapter on linear algebra topics.
- Strengthening the presentation of review topics, including hyperbolic functions and complex numbers.
- Inserting instruction and exercises using the computer algebra system Maxima or SageMath.
- Adding applications in healthcare, chemistry and pharmacy.

Further content was adapted from these open-source resources:

*Linear Algebra*by Jim Hefferon.*Community Calculus*by David Guichard et al.*Precalculus*by Carl Stitz and Jeff Zeager.*Elementary Differential Equations*by William F. Trench*.*

The following are available for this resource:

- Download the textbook as a single PDF file Fall 2017 edition (typeset on July 6, 2017)
- Download the LaTeX source folder as a zipped file (9.0 MB)

**Includes**: Syllabus, progress packet, easy guide to Sage for Differential Equations, example solutions using Maxima, videos

# Discover Linear Algebra

Description: This text is written by Jeremy Sylvestre of the University of Alberta.

*Discover Linear Algebra* is an open-access linear algebra textbook that uses a discovery-based approach to introduce students to this beautiful subject. The philosophy of this treatment is to allow the undeniable core ideas and patterns of linear algebra to reveal themselves to the student.

Each chapter begins with a set of *guided-discovery* activities suitable for use as in-class group activities, as pre-class preparatory explorations, or for self-study. The exposition in the remainder of each chapter reflects and expands upon these introductory explorations, beginning with an informal *Concepts* section, followed by a section of *Examples*, and ending with a more formal *Theory* section of theorems and proofs. Please see the book’s preface (two-semester version; one-semester version) for a little more detail on the organization and pedagogical approach of the book.

*Discover Linear Algebra* is free as in “freedom” — released under the GNU Free Documentation License (GFDL), you are free to use, copy, redistribute, and/or modify this textbook. (Though that freedom comes with some responsibilities; see the full text of the GFDL, included as an appendix to the textbook.) If you wish to make use of this work under a different license, please contact the author using the contact info below. The following are available:

- Discover Linear Algebra – Winter 2019 Edition: Two-semester version
- Discover Linear Algebra: A First Course in Linear Algebra – Winter 2020 Edition: One-semester version
- Github repository for the PreTeXt document source code
**:**github.com/jjrsylvestre/dla - Author contact: Jeremy Sylvestre, University of Alberta, Augustana Campus, jeremy.sylvestre@ualberta.ca

**Includes**: Guided discovery activities, HTML version.

# Discrete Mathematics: An Open Introduction

Description:* Discrete Mathematics: An Open Introduction* is a free, open source textbook appropriate for a first or second year undergraduate course for math majors, especially those who will go on to teach. Since Spring 2013, the book has been used as the primary textbook or a supplemental resource at more than 75 colleges and universities around the world (see the partial adoptions list). The text is endorsed by the American Institute of Mathematics’ Open Textbook Initiative and is well reviewed on the Open Textbook Library. The author of this text is Ocar Levin.

This 3rd edition brings many improvements, including nearly 100 new exercises, a new section on trees in the graph theory chapter, and improved exposition throughout. Previous editions will continue to be available indefinitely. A few times a year, the text is updated with a new “printing” to correct errors. See the errata list for more information.

**New for Fall 2019:** Online homework sets are available through Edfinity or as WeBWorK sets from the author. Additional exercises have been added since Spring 2020.

**Get the book**

The entire book is available for free as an interactive online ebook. This should work well on all screen sizes, including smart phones. Hints and solutions to examples and exercises are hidden but easily revealed by clicking on their links. Some exercises also allow you to enter and check your work, so you can try multiple times without spoiling the answer.

For offline use, a free pdf version, suitable for reading on a tablet or computer, is available for download. This should be searchable and easy to navigate using embedded links. Hints and solutions (when available) can be accessed by clicking on the exercise number, and clicking on the number of the hint or solution will bring you back to the exercise.

If you prefer a physical copy, an inexpensive print version of the text is available on Amazon. This should be cheaper than printing the entire book and binding it yourself. Page numbers match the pdf version.

**Includes: **The following come with this text:

- 473 exercises, including 275 with solutions and another 109 with hints. Exercises range from easy to quite involved, with many problems suitable for homework.
*Investigate!*activities throughout the text to support active, inquiry based learning.- A full index and list of symbols.
- Consistent and helpful page layout and formatting (i.e., examples are easy to identify, important definitions and theorems in boxes, etc.).

## First-Semester Abstract Algebra: A Structural Approach

Author: Jessica K. Sklar

Description: ** A note on algebra:** At its most basic level, abstract algebra is the study of structures. Just as an architect may examine buildings or an anthropologist societal hierarchies, an algebraist explores the nature of sets equipped with binary operations that satisfy certain properties. While these structures may not seem at first to be very important, they are at the heart of most, if not all, mathematical endeavors. On an elemental level, they allow us to solve systems of equations; on a more global-level, they are behind some of our most important cryptographic systems. We even use them implicitly when telling time!

Our focus in this book is the study of algebraic structures called *groups*. Along the way, we will explore rigorous mathematical notions of similarity and difference: When can we consider two objects to be more or less “the same”? When are they fundamentally different? For instance, consider two houses that have exactly the same construction, but are painted different colors. Are they the same house? No. But viewed structurally (as opposed to aesthetically) they are the same. This means that if we know certain information about one of the houses (say, how far the bathroom is from the kitchen) we know the same information about the other house. However, knowing that the first house is painted yellow does not tell us anything about the second house’s color. We explore an analogous idea in mathematics, namely, the concept of *isomorphism*.

Throughout, we provide readers with many mathematical proofs, as well as specific examples demonstrating more general ideas.

# FREE: Federal Resources for Educational Excellence

# Guide to Cultivating Complex Analysis

The purpose of this book is to teach a one-semester graduate course in complex analysis for incoming graduate students. It was created to teach Math 5283 at Oklahoma State University. It is a natural first semester in a two semester sequence where the second semester could be several complex variables or perhaps harmonic analysis. It could perhaps be used for a more elementary two-semester sequence if the appendix (metric spaces, some basic analysis) is covered first, and all the optional bits of the main text are also covered. The text was originally developed by Jiri Lebl in 2020. The following are available for download:

- Download the book as pdf (Version 1.0, September 10th, 2020, 304 pages)
- Latex source code hosted on GitHub.

**Includes**: Exercises (600) in the text.

# Introductory Statistics

# Introductory Statistics: Saylor

This peer-reviewed resource introduces statistical concepts, including: descriptive statistics, basic concepts of probability, discrete random variables, continuous random variables, sampling distributions, estimation, testing hypotheses, two-sample problems, correlation and regression, and chi-square and f-tests.

**Includes**: exercises and answers.

# Khan Academy

Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom.

# Linear Algebra

This textbook has been positively reviewed by four math professors from American universities and the Mathematical Association of America.

**Includes**: exercises, solutions, a lab manual, lecture slides, and includes applications of linear algebra.

# Linear Algebra

A free linear algebra textbook and online resource written by David Cherney, Tom Denton, Rohit Thomas and Andrew Waldron. Edited by Katrina Glaeser and Travis Scrimshaw.

This “textbook” (+videos+WeBWorKs) is suitable for a sophomore level linear algebra course taught in about twenty-five lectures. It is designed both for engineering and science majors, but has enough abstraction to be useful for potential math majors. Our goal in writing it was to produce students who can perform computations with linear systems and also understand the concepts behind these computations. The text was last updated in 2016. The following are downloadable for this text:

- the entire book can be downloaded as a single pdf file of 1.4 MB and 410 pages. (Last update: August 24, 2016)
- The LaTeX source files can be downloaded as a single tarball. (Last update: January 25, 2019)
- The WeBWorK files can be downloaded as a single tarball. (Last update: August 24, 2016)

**Includes**: There are Youtube videos linked to content throughout the book, problems at the end of each chapter, online homework using the WebWork system.

# Linear Algebra with Applications: Lyryx

Description: The first version of this textbook was published in 1986 as a traditional textbook. The current version from 2018 was published as an open educational resource for a traditional or advanced introduction to the topics within linear algebra.

**Includes**: instructor resources: solution manual to exercises and lecture slides.

# Math Explorer Activity Database

# Matrix Algebra with Computational Applications

Description: Matrix Algebra with Computational Applications is a collection of Open Educational Resource (OER) materials designed to introduce students to the use of Linear Algebra to solve real world problems. These materials were developed specifically for students and instructors working in a “flipped classroom” model that emphasizes hands-on problem solving activities during class meetings, with students watching lectures and completing readings and assignments outside of the classroom. The materials are organized into a semester long course with “pre-class” and “in-class” assignments. The “pre-class” assignments include readings, video lectures and coding projects (in Python), which students are expected to complete before attending class. The in-class assignments consist of hands-on individual and group activities intended to be completed during class. These in-class activities are supervised by the instructors, who actively answer questions and help guide the students in achieving the learning goals for the course.

# MIT OpenCourseWare: Mathematics

# Open Course Library

# OpenStax (aka Connexions) Mathematics and Statistics

# PhET: Math

# UBC Calculus Textbooks

UBC has created its own sequence of open calculus texts to be used in the following courses:

- CLP-1 Differential Calculus, also available online using PreTeXt.
- CLP-2 Integral Calculus, also available online using PreTeXt.
- CLP-3 Multivariable Calculus, also available online using PreTeXt.
- CLP-4 Vector Calculus, also available online using PreTeXt.

The latex source from the first two of these texts has been made available from github (source for CLP-1, CLP-2, CLP-3 , CLP-4). Each of the texts also includes a problem book.

**Includes**: Problem book for each of the texts. Exercises and solutions included many of which were taken from old exams, midterm tests and quizzes.

# Understanding Linear Algebra

Description: This is an open access textbook written by David Austin.

An HTML version and PDF version are now available at no cost. Besides being freely available, David Austin intends to make the PreTeXt source code available on GitHub in the near future.

**Includes**: Activities for teaching Linear Algebra with downloadable latex source files, A blog for the text, embedded Sage calculations in HTML version

# Vector Calculus

Description: This text was written by Michael Corral in 2013. This is a text on elementary multivariable calculus, designed for students who have completed courses in single-variable calculus. The traditional topics are covered: basic vector algebra; lines, planes and surfaces; vector-valued functions; functions of 2 or 3 variables; partial derivatives; optimization; multiple integrals; line and surface integrals.

The book also includes discussion of numerical methods: Newton’s method for optimization, and the Monte Carlo method for evaluating multiple integrals. There is a section dealing with applications to probability. Appendices include a proof of the right-hand rule for the cross product, and a short tutorial on using Gnuplot for graphing functions of 2 variables. The following are available for download:

- complete book can be downloaded – the latest version (2013-05-21)
- LaTeX source code is available from here
- Java programs and source code for Newton’s algorithm (Ch. 2) and the Monte Carlo method (Ch. 3)
- Sage equivalents of the above Java programs (2012-02-13)
- Matlab/Octave equivalents of the above Java programs provided by Prof. Benson Muite (University of Michigan) in 2013-05-21.

**Includes**: Exercises with selected answers, some Java, Sage, Matlab/Octave programs illustrating specific concepts.