Math Elementary Linear Algebra Spring Edion, University of Lethbridge (This edion is essenally unchanged from the Spring edion.) Editor: Sean Fitzpatrick Department of Mathemacs and Computer Science University of Lethbridge Contribung Textbooks Precalculus Version bπc = 3 Carl Stz and Jeff Zeager www.stz-zeager.com Fundamentals of Matrix Algebra Third Edion, Version . Gregory Hartman www.vmi.edu APEX Calculus Version . Gregory Hartman et al apexcalculus.com Chapters and Copyright © Carl Stz and Jeff Zeager Licensed under the Creave Commons Aribuon- Noncommercial-ShareAlike . Unported Public License Chapter Copyright © Gregory Hartman Licensed under the Creave Commons Aribuon- Noncommercial . Internaonal Public License Chapters - Copyright © Gregory Hartman, except as noted below. Licensed under the Creave Commons Aribuon- Noncommercial . United States License This version of the text was assembled and edited by Sean Fitzpatrick, Uni- versity of Lethbridge, July-August, , most recently updated January . In addion to minor changes and addions throughout the text, supplemen- tary content for this text was wrien by Sean Fitzpatrick, including almost all of Secons ., ., ., ., and . (exercises in Secon . were contributed by Jana Archibald). Some eding and addions also took place in other secons, including .-., ., ., ., ., ., and .. This work is licensed under the Creave Commons Aribuon-Noncommercial- ShareAlike . Internaonal Public License Contents Table of Contents iii Preface v The Real Numbers . Some Basic Set Theory Noons .................. . Real Number Arithmec ..................... . The Cartesian Coordinate Plane .................. The Complex Numbers . Complex Numbers ......................... . Polar Coordinates ......................... . The Polar Form of Complex Numbers ............... Vectors . Introducon to Cartesian Coordinates in Space ......... . An Introducon to Vectors .................... . The Dot Product .......................... . The Cross Product ......................... . Lines ................................ . Planes ............................... . The Vector Space Rn ....................... Systems of Linear Equaons . Introducon to Linear Equaons ................. . Using Matrices To Solve Systems of Linear Equaons ...... . Elementary Row Operaons and Gaussian Eliminaon ...... . Existence and Uniqueness of Soluons .............. . Applicaons of Linear Systems .................. . Vector Soluons to Linear Systems ................ Matrix Algebra . Matrix Addion and Scalar Mulplicaon ............ . Matrix Mulplicaon ....................... . Solving Matrix Equaons AX = B ................ . The Matrix Inverse ........................ . Properes of the Matrix Inverse ................. . Elementary Matrices ....................... Matrix Transformaons . Matrix Transformaons ...................... . Properes of Linear Transformaons ............... . Subspaces of Rn ......................... Contents . Null Space and Column Space ................... Operaons on Matrices . The Matrix Transpose ....................... . The Matrix Trace ......................... . The Determinant ......................... . Properes of the Determinant .................. . Applicaons of the Determinant ................. Eigenvalues and Eigenvectors . Eigenvalues and Eigenvectors ................... . Properes of Eigenvalues and Eigenvectors ........... . Eigenvalues and Diagonalizaon ................. A Answers To Selected Problems A. Index A. iv Preface This is a textbook on Linear Algebra, designed to be compable with the curriculum for the course Math at the University of Lethbridge. As this is a first-year course taken primarily by non-majors, the focus of the text (and the course) is primarily computaonal, and as a result, much of the theory that common to many first courses in linear algebra is not included. The book does, however, contain a few topics not always found in a linear algebra course (but required for Math ), including vector geometry and com- plex numbers. The book is free, in every sense of the word. There is no cost to the student, unless one wants a hard copy, in which case the only cost is that of having it printed. The text is also free in less tangible, but equally important ways. The book is licensed under a Creave Commons Public License, which allows you to share the book with others, and even make and distribute copies, as long as this is not done for financial gain. The book is also open source: all of the code and figures used to generate the text (over , files in all) are available online and can be downloaded and edited by anyone wishing to create their own version of the book. The book has a few quirks compared to other texts. There is an extensive treatment of the arithmec of complex numbers, since this is required in the Math course. The treatment of systems of equaons also appears quite late in the textbook. In my experience, when the course begins with systems of equaons, many students tune out early, either because the material is too easy, or too boring, and aren’t able to get back up to speed once the level of difficulty picks up. Having some challenging material early on provides some incenve to do the homework and seek help from the beginning. Since Math does not align well with the standard linear algebra cur- riculum found in many American universies, it was not possible to work with any one pre-exisng open textbook. Instead, this book is an amalgamaon of three texts, as specified on the copyright page, along with a substanal chunk of content that I’ve added myself. As a result, you’ll find that there are slight differences in the format of some chapters. The treatment of standard topics like basis and dimension is quite limited, due to the nature of the course. Arguments are provided in support of most theorems in the text, but there aren’t too many formal proofs. Some topics that are somemes taught in Math also receive a limited treatment here. Eigenvalues and eigenvectors are covered, but diagonalizaon is not. (Update for the Spring edion: a secon on diagonalizaon has been added.) The coverage of orthogonality is limited to the shortest distance applicaons in the chapter on vector geometry. In some cases these omissions are due to me constraints, both in the preparaon of the textbook and in the delivery of the material in the classroom. In some cases (such as the Gram-Schmidt procedure and orthogonal diagonalizaon), there was a concious decision to omit material that more rightly belongs in a second course in linear algebra. With any luck, the book will connue to evolve as I and other instructors use it. The book is a work in progress, and is bound to have some failings. If you encounter any errors as you use the book (or if you don’t like how something is presented, or if you think more exercises of a certain type are needed, etc.) please let me know, and I’ll do my best to make the changes. Contents Changes in the Spring edion For Spring , the course is being taught by Jana Archibald, who asked that we re-instute the chapter on linear transformaons. She also contributed a (previously missing) set of problems for the secon on elementary matrices. Changes in the Fall edion For Fall I’ve reorganized the textbook based on the input of Habiba Kadiri, since she will be teaching Math this semester. The main change is to bring the material on systems of equaons a bit earlier in the textbook, placing it prior to the chapter on matrices. A few adjustments in wring were needed to accommodate this change. Some of the theorecal content has been omied from the chapter on vectors in Rn. The material on linear transformaons has been moved to the end of the textbook as an oponal chapter, to be covered if me permits. I’ve moved the material on null space and column space into its own secon at the end of this chapter. This move includes relocang or re- producing the relevant examples from the secon on vector soluons to linear systems. (Time constraints kept me from creang new examples, so there are some examples in Secon . that have been re-wrien to remove the refer- ences to null space and column space, and the same examples reappear in their original context at the end of the book.) I have also wrien a new secon, on elementary matrices. In the previous edion the only menon of this topic was in a marginal note. Changes in the Spring edion During the inaugural Fall run for this textbook, I came across a number of errors and typos that have been corrected. Most of these were formang errors, but there were a few incorrect answers in the back, and one case in Sec- on . where I somehow used the columns from the wrong matrix in a column space example! In Chapter I’ve added formal definions for addion and mulplicaon of complex numbers, and the presentaon in Secon . has been streamlined. Since the polar coordinate representaon of a complex number never has r < 0, I’ve removed the material (needed for calculus) involving polar coordinates with r < 0, as well as the material (also needed for calculus) on converng equaons of curves in the plane from rectangular to polar coordinates. Replacing the “cis” notaon with Euler’s exponenal notaon remains on the to-do list. In Chapter , I’ve added the standard definion of parallel vectors for linear algebra, and changed the original definion to a theorem. In Chapter , I’ve added addional examples and exercises on span and linear independence in Secon ., and two addional examples in Secon . on working