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Linear Algebra I Skills, Concepts and Applications

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Linear Algebra I Skills, Concepts and Applications Gregg Waterman Oregon Institute of Technology c 2016 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International license. The essence of the license is that You are free to: Share - copy and redistribute the material in any medium or format • Adapt - remix, transform, and build upon the material for any purpose, even commercially. • The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms: Attribution - You must give appropriate credit, provide a link to the license, and indicate if changes • were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions - You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Notices: You do not have to comply with the license for elements of the material in the public domain or where your use is permitted by an applicable exception or limitation. No warranties are given. The license may not give you all of the permissions necessary for your intended use. For example, other rights such as publicity, privacy, or moral rights may limit how you use the material. For any reuse or distribution, you must make clear to others the license terms of this work. The best way to do this is with a link to the web page below. To view a full copy of this license, visit https://creativecommons.org/licenses/by/4.0/legalcode. Contents 1 Systems of Linear Equations 1 1.1 Linear Equations and Systems of Linear Equations . ......... 2 1.2 Curve Fitting, Temperature Equilibrium and Electric Circuits........... 7 1.3 SolvingSystemsofLinearEquations . ..... 13 1.4 SolvingWithMatrices .............................. 17 1.5 “WhenThingsGoWrong”............................. 25 1.6 BacktoApplications............................... 31 1.7 Chapter1Exercises ................................ 35 2 Euclidean Space and Vectors 40 2.1 EuclideanSpace .................................. 41 2.2 IntroductiontoVectors . .. .. .. 45 2.3 Addition and Scalar Multiplication of Vectors, Linear Combinations . 48 2.4 LinearCombinationFormofaSystem. .... 54 2.5 SetsofVectors................................... 58 2.6 VectorEquationsofLinesandPlanes . ..... 62 2.7 Interpreting Solutions to Systems of Linear Equations . ............ 68 2.8 The Dot Product of Vectors, Projections . ...... 72 3 Matrices and Vectors 76 3.1 IntroductiontoMatrices. ... 78 3.2 MatrixTimesaVector............................... 82 3.3 Actions of Matrices on Vectors: Transformations in R2 ............. 90 3.4 MultiplyingMatrices .............................. .. 94 3.5 InverseMatrices .................................. 102 3.6 Determinants and Systems of Equations . ...... 108 3.7 Applications: Transformation Matrices, Graph Theory . ............ 115 4 Vector Spaces and Subspaces 121 4.1 SpanofaSetofVectors.............................. 122 4.2 ClosureofaSetUnderanOperation . 128 4.3 Vector Spaces and Subspaces . 131 4.4 ColumnSpaceandNullSpaceofaMatrix . 138 4.5 Least Squares Solutions to Inconsistent Systems . .......... 142 4.6 LinearIndependence ............................... 147 4.7 BasesofSubspaces,Dimension . 155 4.8 Bases for the Column Space and Null Space of a Matrix . ....... 160 4.9 SolutionstoSystemsofEquations. ..... 165 4.10Chapter4Exercises ............................... 168 5 Linear Transformations 170 5.1 TransformationsofVectors . 171 5.2 LinearTransformations. 176 5.3 Linear Transformations and Matrices . ...... 183 5.4 CompositionsofTransformations . ..... 186 5.5 Transformations and Homogeneous Coordinates . ........ 190 5.6 An Introduction to Eigenvalues and Eigenvectors . .......... 197 5.7 Finding Eigenvalues and Eigenvectors . ....... 202 i 5.8 DiagonalizationofMatrices . .... 211 5.9 Solving Systems of Differential Equations . ........ 214 A Index of Symbols 216 B Solutions to Exercises 218 B.1 Chapter1Solutions ................................ 218 B.2 Chapter2Solutions ................................ 219 B.3 Chapter3Solutions ................................ 219 B.4 Chapter4Solutions ................................ 220 B.5 Chapter5Solutions ................................ 223 B.6 Chapter6Solutions ................................ 223 B.7 Chapter7Solutions ................................ 224 B.8 Chapter8Solutions ................................ 226 B.9 Chapter9Solutions ................................ 228 B.10Chapter10Solutions. .. .. 230 C Solutions to Exercises 232 C.1 Chapter1Solutions ................................ 232 C.2 Chapter2Solutions ................................ 234 C.3 Chapter3Solutions ................................ 241 C.4 Chapter4Solutions ................................ 247 C.5 Chapter5Solutions ................................ 250 ii 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) Set up a system of linear equations to find coefficients of a line or poly- nomial through a given set of points, or to model flow in a network or equilibrium temperatures in a solid object. (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) Give the coefficient matrix and augmented matrix for a system of equa- tions. (f) Determine whether a matrix is in row-echelon form. Perform, by hand, elementary row operations to reduce a matrix to row-echelon form. (g) Determine whether a matrix is in reduced row-echelon form. Use tech- nology to reduce a matrix to reduced row-echelon form. (h) For a system of equations having a unique solution, determine the so- lution from either the row-echelon form or reduced row-echelon form of the augmented matrix for the system. (i) Use a calculator to solve a system of linear equations having a unique solution. (j) Given the row-echelon or reduced row-echelon form of an augmented matrix for a system of equations, determine the leading variables and free variables of the system. (k) Given the row-echelon or reduced row-echelon form for a system of equa- tions: Determine whether the system has a unique solution, and give the • solution if it does. If the system does not have a unique solution, determine whether it • is inconsistent (no solution) or dependent (infinitely many solutions). If the system is dependent, give the general form of a solution and • give some particular solutions. (l) Use systems of equations to solve network problems. 1 1.1 Linear Equations and Systems of Linear Equations Performance Criteria: 1. (a) Determine whether an equation in n unknowns is linear. (b) Set up a system of linear equations to find coefficients of a line or poly- nomial through a given set of points, or to model flow in a network or equilibrium temperatures in a solid object. (c) Determine whether an n-tuple is a solution to a linear equation or a system of linear equations. Linear Equations and Their Solutions It is natural to begin our study of linear algebra with the process of solving systems of linear equations, and applications of such systems. Definition 1.1.1: A linear equation in n unknowns is an equation that can be put in the form a x + a x + a x + + a x = b, (1) 1 1 2 2 3 3 · · · n n where a1, a2, ..., an and b are known constants and x1, x2, ... , xn are unknown values. A solution to a linear equation is a collection of values for the unknowns that makes the equation true. Example 1.1(a): Which of the equations ⋄ y = 2 x + 4 y = 16t2 + 61t + 7 5.3x + 7.2y + 1.4z = 16.9 − 3 − a x + a x + + a x = b , 41 1 42 2 · · · 4n n 4 where a11, a12, ..., a1n, b1 are all known numbers, are linear equations? 2 Solution: The first equation can be rewritten as 3 x + y = 4, so it is a linear equation. The second equation can be written 16t2 + 61t y = 4, but the t2 prevents this from being a − − − linear equation. The third and fourth equations are in exactly the form (1), so they are linear. A few comments are in order: For the third equation above we see that • 5.3(1) + 7.2(2) + 1.4( 2) = 16.9, − so x = 1, y = 2 and z = 2 is a solution to 5.3x + 7.2y + 1.4z = 16.9. To save writing we − usually write such a solution as (1, 2, 2), a form you are likely familiar with. − 2 The equation y = 2 x + 4 can also be rewritten as 2x + 3y = 12 instead of 2 x + y = 4. An • − 3 3 (x,y) pair that is a solution to any one of the forms is also a solution to the other two (and any pair that is NOT a solution to any one of them will not be a solution to the other two either). We can multiply or divide both sides of a linear equation by a value in order to make it easier to work with, if we wish. Although you may have used x and y, or x, y and z as the unknown quantities in the • past, like in the third equation above, we will often use x1, x2, ..., xn instead. Thus the third equation could be written 5.3x1 + 7.2x2 + 1.4x3 = 15.9, which is equivalent to the fourth equation with a41 = 5.3, a42 = 7.2, a43 = 1.4 and b4 = 15.9. One obvious advantage to using the letter a for all of the numbers is that we don’t have to fret about what letters to use, and there is no danger of running out of letters! You will eventually see that there is also a very good mathematical reason for using just x (or some other single letter), with subscripts denoting different values. It is important that you easily recognize the form (1) from the definition of a linear equation. Soon we will be interested in similar equations, but of the form number vector + number vector + + number vector = vector. · · · · · · We now move on to the concept that forms the beginning of our study of linear algebra: Definition 1.1.2: A system of linear equations is a set of linear equations containing the same unknowns. (Not every equation needs to contain every un- known.) A solution to a system of linear equations is a collection of values for the unknowns that makes every equation of the system true.
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