LINEAR ALGEBRA - Mathematics & Statistics

LINEAR ALGEBRA

Paul Dawkins

Linear Algebra

Table of Contents

Preface ............................................................................................................................................ ii Outline ........................................................................................................................................... iii Systems of Equations and Matrices ............................................................................................. 1

Introduction ................................................................................................................................................ 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................................................................................................................................ 90 Introduction .............................................................................................................................................. 90 The Determinant Function ....................................................................................................................... 91 Properties of Determinants......................................................................................................................100 The Method of Cofactors ........................................................................................................................107 Using Row Reduction To Compute Determinants ..................................................................................115 Cramer's Rule .........................................................................................................................................122 Euclidean n-Space ..................................................................................................................... 125 Introduction .............................................................................................................................................125 Vectors ....................................................................................................................................................126 Dot Product & Cross Product..................................................................................................................140 Euclidean n-Space...................................................................................................................................154 Linear Transformations ...........................................................................................................................163 Examples of Linear Transformations ......................................................................................................173 Vector Spaces ............................................................................................................................. 181 Introduction .............................................................................................................................................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 .................................................................................................. 305 Introduction .............................................................................................................................................305 Review of Determinants..........................................................................................................................306 Eigenvalues and Eigenvectors.................................................................................................................315 Diagonalization .......................................................................................................................................331

? 2007 Paul Dawkins

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Linear Algebra

Pref ace

Here 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 wasn't 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 don't have time in class to work all of the problems in the notes and so you will find that some sections contain problems that weren't 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 can't anticipate all the questions. Sometimes a very good question gets asked in class that leads to insights that I've 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

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Linear Algebra

Outline

Here is a listing and brief description of the material in this set of notes.

Systems of Equations and Matrices Systems of Equations ? In this section we'll introduce most of the basic topics that we'll need in order to solve systems of equations including augmented matrices and row operations. Solving Systems of Equations ? Here we will look at the Gaussian Elimination and Gauss-Jordan Method of solving systems of equations. Matrices ? We will introduce many of the basic ideas and properties involved in the study of matrices. Matrix Arithmetic & Operations ? In this section we'll take a look at matrix addition, subtraction and multiplication. We'll also take a quick look at the transpose and trace of a matrix. Properties of Matrix Arithmetic ? We will take a more in depth look at many of the properties of matrix arithmetic and the transpose. Inverse Matrices and Elementary Matrices ? Here we'll define the inverse and take a look at some of its properties. We'll also introduce the idea of Elementary Matrices. Finding Inverse Matrices ? In this section we'll develop a method for finding inverse matrices. Special Matrices ? We will introduce Diagonal, Triangular and Symmetric matrices in this section. LU-Decompositions ? In this section we'll introduce the LU-Decomposition a way of "factoring" certain kinds of matrices. Systems Revisited ? Here we will revisit solving systems of equations. We will take a look at how inverse matrices and LU-Decompositions can help with the solution process. We'll also take a look at a couple of other ideas in the solution of systems of equations.

Determinants The Determinant Function ? We will give the formal definition of the determinant in this section. We'll also give formulas for computing determinants

of 2? 2 and 3? 3 matrices.

Properties of Determinants ? Here we will take a look at quite a few properties of the determinant function. Included are formulas for determinants of triangular matrices. The Method of Cofactors ? In this section we'll take a look at the first of two methods form computing determinants of general matrices. Using Row Reduction to Find Determinants ? Here we will take a look at the second method for computing determinants in general. Cramer's Rule ? We will take a look at yet another method for solving systems. This method will involve the use of determinants.

Euclidean n-space Vectors ? In this section we'll introduce vectors in 2-space and 3-space as well as some of the important ideas about them.

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Linear Algebra

Dot Product & Cross Product ? Here we'll look at the dot product and the cross product, two important products for vectors. We'll also take a look at an application of the dot product. Euclidean n-Space ? We'll introduce the idea of Euclidean n-space in this section and extend many of the ideas of the previous two sections. Linear Transformations ? In this section we'll introduce the topic of linear transformations and look at many of their properties. Examples of Linear Transformations ? We'll take a look at quite a few examples of linear transformations in this section.

Vector Spaces Vector Spaces ? In this section we'll formally define vectors and vector spaces. Subspaces ? Here we will be looking at vector spaces that live inside of other vector spaces. Span ? The concept of the span of a set of vectors will be investigated in this section. Linear Independence ? Here we will take a look at what it means for a set of vectors to be linearly independent or linearly dependent. Basis and Dimension ? We'll be looking at the idea of a set of basis vectors and the dimension of a vector space. Change of Basis ? In this section we will see how to change the set of basis vectors for a vector space. Fundamental Subspaces ? Here we will take a look at some of the fundamental subspaces of a matrix, including the row space, column space and null space. Inner Product Spaces ? We will be looking at a special kind of vector spaces in this section as well as define the inner product. Orthonormal Basis ? In this section we will develop and use the Gram-Schmidt process for constructing an orthogonal/orthonormal basis for an inner product space. Least Squares ? In this section we'll take a look at an application of some of the ideas that we will be discussing in this chapter. QR-Decomposition ? Here we will take a look at the QR-Decomposition for a matrix and how it can be used in the least squares process. Orthogonal Matrices ? We will take a look at a special kind of matrix, the orthogonal matrix, in this section.

Eigenvalues and Eigenvectors Review of Determinants ? In this section we'll do a quick review of determinants. Eigenvalues and Eigenvectors ? Here we will take a look at the main section in this chapter. We'll be looking at the concept of Eigenvalues and Eigenvectors. Diagonalization ? We'll be looking at diagonalizable matrices in this section.

? 2007 Paul Dawkins

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