paul's linear algebra book

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LINEAR ALGEBRAPaul Dawkins

Linear Algebra

Table of ContentsPreface............................................................................................................................................. ii Outline............................................................................................................................................ iii Systems of Equations and Matrices.............................................................................................. 1Introduction ................................................................................................................................................ 1 Systems of Equations ................................................................................................................................. 3 Solving Systems of Equations .................................................................................................................. 15 Matrices .................................................................................................................................................... 27 Matrix Arithmetic & Operations .............................................................................................................. 33 Properties of Matrix Arithmetic and the Transpose ................................................................................. 45 Inverse Matrices and Elementary Matrices .............................................................................................. 50 Finding Inverse Matrices .......................................................................................................................... 59 Special Matrices ....................................................................................................................................... 68 LU-Decomposition ................................................................................................................................... 75 Systems Revisited .................................................................................................................................... 81

Determinants ................................................................................................................................ 90Introduction .............................................................................................................................................. 90 The Determinant Function ....................................................................................................................... 91 Properties of Determinants ..................................................................................................................... 100 The Method of Cofactors ....................................................................................................................... 107 Using Row Reduction To Compute Determinants ................................................................................. 115 Cramers Rule ........................................................................................................................................ 122

Euclidean n-Space ...................................................................................................................... 125Introduction ............................................................................................................................................ 125 Vectors ................................................................................................................................................... 126 Dot Product & Cross Product ................................................................................................................. 140 Euclidean n-Space .................................................................................................................................. 154 Linear Transformations .......................................................................................................................... 163 Examples of Linear Transformations ..................................................................................................... 173

Vector Spaces ............................................................................................................................. 181Introduction ............................................................................................................................................ 181 Vector Spaces ......................................................................................................................................... 183 Subspaces ............................................................................................................................................... 193 Span ........................................................................................................................................................ 203 Linear Independence .............................................................................................................................. 212 Basis and Dimension .............................................................................................................................. 223 Change of Basis...................................................................................................................................... 239 Fundamental Subspaces ......................................................................................................................... 252 Inner Product Spaces .............................................................................................................................. 263 Orthonormal Basis ................................................................................................................................. 271 Least Squares ......................................................................................................................................... 283 QR-Decomposition ................................................................................................................................ 291 Orthogonal Matrices ............................................................................................................................... 299

Eigenvalues and Eigenvectors ................................................................................................... 305Introduction ............................................................................................................................................ 305 Review of Determinants ......................................................................................................................... 306 Eigenvalues and Eigenvectors ................................................................................................................ 315 Diagonalization ...................................................................................................................................... 331

2007 Paul Dawkins

i

http://tutorial.math.lamar.edu/terms.aspx

Linear Algebra

PrefaceHere are my online notes for my Linear Algebra course that I teach here at Lamar University. Despite the fact that these are my class notes, they should be accessible to anyone wanting to learn Linear Algebra or needing a refresher. These notes do assume that the reader has a good working knowledge of basic Algebra. This set of notes is fairly self contained but there is enough Algebra type problems (arithmetic and occasionally solving equations) that can show up that not having a good background in Algebra can cause the occasional problem. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. 1. Because I wanted to make this a fairly complete set of notes for anyone wanting to learn Linear Algebra I have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here. You will need to find one of your fellow class mates to see if there is something in these notes that wasnt covered in class. 2. In general I try to work problems in class that are different from my notes. However, with a Linear Algebra course while I can make up the problems off the top of my head there is no guarantee that they will work out nicely or the way I want them to. So, because of that my class work will tend to follow these notes fairly close as far as worked problems go. With that being said I will, on occasion, work problems off the top of my head when I can to provide more examples than just those in my notes. Also, I often dont have time in class to work all of the problems in the notes and so you will find that some sections contain problems that werent worked in class due to time restrictions. 3. Sometimes questions in class will lead down paths that are not covered here. I try to anticipate as many of the questions as possible in writing these notes up, but the reality is that I cant anticipate all the questions. Sometimes a very good question gets asked in class that leads to insights that Ive not included here. You should always talk to someone who was in class on the day you missed and compare these notes to their notes and see what the differences are. 4. This is somewhat related to the previous three items, but is important enough to merit its own item. THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!! Using these notes as a substitute for class is liable to get you in trouble. As already noted not everything in these notes is covered in class and often material or insights not in these notes is covered in class.

2007 Paul Dawkins

ii

http://tutorial.math.lamar.edu/terms.aspx

Linear Algebra