Principles of Linear Algebra with Mathematica

Kenneth M. Shiskowski, Karl Frinkle

• Publisher: Wiley
• Year: 2013
• ISBN:
  9781118627266
• Edition: 1
• Topic: Mathematics
• Subtopic: Linear Algebra
• Index: Mathematics

A hands-on introduction to the theoretical and computational aspects of linear algebra using Mathematica®

Many topics in linear algebra are simple, yet computationally intensive, and computer algebra systems such as Mathematica® are essential not only for learning to apply the concepts to computationally challenging problems, but also for visualizing many of the geometric aspects within this field of study. Principles of Linear Algebra with Mathematica uniquely bridges the gap between beginning linear algebra and computational linear algebra that is often encountered in applied settings, and the commands required to solve complex and computationally challenging problems using Mathematica are provided.

The book begins with an introduction to the commands and programming guidelines for working with Mathematica. Next, the authors explore linear systems of equations and matrices, applications of linear systems and matrices, determinants, inverses, and Cramer's rule. Basic linear algebra topics, such as vectors, dot product, cross product, and vector projection are explored, as well as a unique variety of more advanced topics including rotations in space, 'rolling' a circle along a curve, and the TNB Frame. Subsequent chapters feature coverage of linear transformations from Rn to Rm, the geometry of linear and affine transformations, with an exploration of their effect on arclength, area, and volume, least squares fits, and pseudoinverses.

Mathematica is used to enhance concepts and is seamlessly integrated throughout the book through symbolic manipulations, numerical computations, graphics in two and three dimensions, animations, and programming. Each section concludes with standard problems in addition to problems that were specifically designed to be solved with Mathematica, allowing readers to test their comprehension of the presented material. All related Mathematica code is available on a corresponding website, along with solutions to problems and additional topical resources.

Extensively class-tested to ensure an accessible presentation, Principles of Linear Algebra with Mathematica is an excellent book for courses on linear algebra at the undergraduate level. The book is also an ideal reference for students and professionals who would like to gain a further understanding of the use of Mathematica to solve linear algebra problems.

Chapter 1

An Introduction to Mathematica

1.1 The Very Basics

Mathematica is an extremely powerful mathematical software package (or computer algebra system) that incorporates text editing, mathematical computation, and programming as well as 2D and 3D graphics capabilities. You can literally write a complete mathematics textbook using only Mathematica where your book includes all of the text and graphics in one smoothly flowing document. If you have never or only slightly used Mathematica before, then it will take some effort to learn how it works—believe me that it is well worth the time expended for the ability to do mathematically almost anything you can dream of that a computer might be able to do for you. In this introduction to Mathematica, you will see only a fraction of its capabilities, but hopefully enough to get you well on your way in doing 2D and 3D graphics, solving of equations, defining and using functions, lists and matrices, along with some basic mathematical programming.

This chapter discusses the fundamentals of using Mathematica for the novice user. If you are already familiar with Mathematica, you may wish to skip this chapter, although we warn you that to do so would be at your own risk. The new user of Mathematica will find it quite difficult in the beginning, but with practice and patience, you will master all of the basics and in time come to enjoy using Mathematica.

Mathematica files are called notebooks, and in a notebook you can place text along with input commands and their associated outputs which can be literally anything such as graphics, tables or lists, and functions. You can group the material in a notebook into different types of cells that are indicated on the right side of the notebook by brackets. At the top of the notebook you will see the tab Palettes and under it is the Writing Assistant, which will allow you to create new cells and/or modify cells. You can use Writing Assistant to change the font, color, and size of the text in your cells and you can also do this using the Format tab at the top of the notebook. The word processing capabilities of Mathematica are very similar to those of Microsoft Word with

as copy and

as paste,

as delete and copy, which can be pasted elsewhere.

The commands

and

will create Input and Text cells, respectively, after a horizontal line break between cells. Almost all Mathematica cells are Input or Text cells, or Output cells that are created when you activate an Input cell. Input and Output cells are normally in pairs with Output second directly following its Input. You can also create a new cell after a line break by typing in some text where you can control the type of cell you are writing in by using the menu which is open at the upper left of the screen next to the Save (or disk) icon.

Each section or chapter of a notebook file in Mathematica should be created as a section where the first cell of the section is a Subtitle cell that can be created by placing a horizontal bar between or just after a cell and then choosing Subtitle from the pulldown menu at the very top left of the lower ruler at the top of your screen. In order to create a subsection of this section (or chapter), do the same as just described but choose the Subsection from this menu. If you have not already used the Window tab to insert the Toolbar in your notebook, then please do so now. With the Toolbar in place, you can now change the type of cell you are in by using the pulldown menu at its far left. The Ruler can also be inserted into your notebook if you want it from the Window tab. Note that for those of us who like our text in a larger style, Window also has a Magnification feature that is quite ...