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Linear Algebra I Linear Algebra I Gregg Waterman Oregon Institute of Technology c 2016 Gregg Waterman This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. The essence of the license is that You are free: to Share to copy, distribute and transmit the work • to Remix to adapt the work • Under the following conditions: Attribution You must attribute the work in the manner specified by the author (but not in • any way that suggests that they endorse you or your use of the work). Please contact the author at [email protected] to determine how best to make any attribution. Noncommercial You may not use this work for commercial purposes. • Share Alike If you alter, transform, or build upon this work, you may distribute the resulting • work only under the same or similar license to this one. With the understanding that: Waiver Any of the above conditions can be waived if you get permission from the copyright • holder. 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Contents 0 Introduction 1 1 Systems of Linear Equations 3 1.1 Linear Equations and Systems of Linear Equations . ............. 4 1.2 TheAdditionMethod ............................... .... 5 1.3 SolvingWithMatrices ............................. ...... 9 1.4 Applications: Curve Fitting and Temperature Equilibrium............... 14 1.5 Chapter1Exercises ............................... ..... 18 2 More on Systems of Linear Equations 23 2.1 “WhenThingsGoWrong” ............................. 24 2.2 Overdetermined and Underdetermined Systems . ............ 30 2.3 Application: Network Analysis . ......... 31 2.4 Approximating Solutions With Iterative Methods . .............. 33 2.5 Chapter2Exercises ............................... ..... 37 3 Euclidean Space and Vectors 43 3.1 EuclideanSpace .................................. 44 3.2 IntroductiontoVectors . ....... 47 3.3 Addition and Scalar Multiplication of Vectors, Linear Combinations . 49 3.4 TheDotProductofVectors,Projections . ........... 55 3.5 Chapter3Exercises ............................... ..... 59 4 Vectors and Systems of Equations 61 4.1 Linear Combination Form of a System . ........ 62 4.2 SetsofVectors ................................... 65 4.3 Vector Equations of Lines and Planes . ......... 69 4.4 Interpreting Solutions to Systems of Linear Equations . ................ 74 4.5 Chapter4Exercises ............................... ..... 77 5 Matrices and Vectors 79 5.1 Introduction.................................... ..... 80 5.2 Multiplying a Matrix Times a Vector . ......... 83 5.3 Actions of Matrices on Vectors: Transformations in R2 ................ 87 5.4 Chapter5Exercises ............................... ..... 94 6 Matrix Multiplication 95 6.1 MultiplyingMatrices .. .. .. .. .. .. .. .. ....... 96 6.2 MoreMultiplyingMatrices . .......102 6.3 InverseMatrices ................................. 106 6.4 Applications of Matrices II: Rotations and Projections,GraphTheory . 110 6.5 Chapter6Exercises ............................... 113 7 Matrices and Systems of Equations 115 7.1 MatrixEquationFormofaSystem . .......116 7.2 Solving a System With An LU-Factorization . 118 7.3 InverseMatricesandSystems . ........121 7.4 Determinants and Matrix Form . .......123 7.5 HomogeneousSystems .............................. 127 7.6 Chapter7Exercises ............................... 128 i 8 Vector Spaces and Subspaces 129 8.1 SpanofaSetofVectors ............................. 130 8.2 ClosureofaSetUnderanOperation . ........135 8.3 VectorSpacesandSubspaces . .......137 8.4 ColumnSpaceandNullSpaceofaMatrix . ........144 8.5 Least Squares Solutions to Inconsistent Systems . ...............147 8.6 Chapter8Exercises ............................... 151 9 Bases of Subspaces 153 9.1 LinearIndependence .............................. 154 9.2 BasesofSubspaces,Dimension . ........160 9.3 Bases for the Column Space and Null Space of a Matrix . ...........164 9.4 SolutionstoSystemsofEquations . .........168 9.5 Chapter9Exercises ............................... 170 10 Linear Transformations 173 10.1 TransformationsofVectors . .........174 10.2 LinearTransformations . ........176 10.3 Compositions of Transformations . ...........183 10.4 Chapter10Exercises. .......186 11 Eigenvalues, Eigenspaces and Diagonalization 187 11.1 An Introduction to Eigenvalues and Eigenvectors . ...............188 11.2 Finding Eigenvalues and Eigenvectors . ............192 11.3 Diagonalization of Matrices . ..........196 11.4 Solving Systems of Differential Equations . ............199 A Index of Symbols 201 B Solutions to Exercises 203 B.1 Chapter1Solutions ............................... 203 B.2 Chapter2Solutions ............................... 204 B.3 Chapter3Solutions ............................... 204 B.4 Chapter4Solutions ............................... 206 B.5 Chapter5Solutions ............................... 207 B.6 Chapter6Solutions ............................... 208 B.7 Chapter7Solutions ............................... 209 B.8 Chapter8Solutions ............................... 211 B.9 Chapter9Solutions ............................... 212 B.10Chapter10Solutions. .......214 ii 0 Introduction This book is an attempt to make the subject of linear algebra as understandable as possible, for a first time student of the subject. I developed the book from a set of notes used to supplement a standard textbook used when I taught the course in the past. At the end of the term I surveyed the students in the class, and the vast majority of them thought that the supplemental notes that I had provided would have been an adequate resource for them to learn the subject. Encouraged by this, I put further work in to correcting errors, adding examples and including material that I had left to the textbook previously. Here is the result, in its fourth edition. You should look at the table of contents and page through the book a bit at first to see how it is organized. Note in particular that each section begins with statements of the performance criteria addressed in that section. Performance criteria are the specific things that you will be expected to do in order to demonstrate your skills and your understanding of concepts. Within each section you will find examples relating to those performance criteria, and at the end of each chapter are exercises for you to practice your skills and test your understanding. It is my belief that you will likely forget many of the details of this course after you leave it, so it might seem that developing the ability to perform the tasks outlined in the performance criteria is in vain. That causes me no dismay, nor should it you! My first goal is for you to learn the material well enough that if you are required to recall or relearn it in other courses you can do so easily and quickly, and along the way I hope that you develop an appreciation for the subject of linear algebra. Most of all, my aim is for you to develop your skills beyond their current level in the areas of mathematical reasoning and communication. In support of that, you will find the first (well, “zeroth”!) outcome and its associated performance criteria below. Outcomes are statements of goals that are perhaps a bit nebulous, and difficult to measure directly. The performance criteria give specific, measurable tasks by which success in the overarching outcome can be determined. Learning Outcome: 0. Use the subject of linear algebra to develop sophistication in understanding of mathematical concepts and connections, and in the communication of that understanding. Performance Criteria: (a) Apply skills and knowledge associated with specific performance cri- teria to problems related to, but not specifically addressed by, those performance criteria. (b) Communicate mathematical ideas clearly by correctly using written English and proper mathematical notation. Enough talk - let’s get to it! Gregg Waterman February 2016 1 2 1 Systems of Linear Equations Learning Outcome: 1. Solve systems of linear equations using Gaussian elimination, use systems of linear equations to solve problems. Performance Criteria: (a) Determine whether an equation in n unknowns is linear. (b) Determine what the solution set to a linear equation represents geo- metrically. (c) Determine whether an n-tuple is a solution to a linear equation or a system of linear equations. (d) Solve a system of two linear equations by the addition method. (e) Find the solution to a system of two linear equations in two unknowns graphically. (f) Give the coefficient matrix and augmented matrix for a system of equations. (g) Determine whether a matrix is in row-echelon form. Perform, by hand, elementary row operations to reduce a matrix to row-echelon form. (h) Determine whether a matrix is in reduced row-echelon form. Use a
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