Interactive Linear Algebra

Transcription

1 Interactive Linear Algebra

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3 Interactive Linear Algebra Joseph Rabinoff Georgia Institute of Technology Dan Margalit Georgia Institute of Technology April 20, 208

4 207 Georgia Institute of Technology Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front- Cover Texts, and no Back-Cover Texts. A copy of the license is included in the appendix entitled GNU Free Documentation License. All trademarks are the registered marks of their respective owners.

5 Contributors to this textbook JOSEPH RABINOFF School of Mathematics Georgia Institute of Technology LARRY ROLEN School of Mathematics Georgia Institute of Technology DAN MARGALIT School of Mathematics Georgia Institute of Technology v

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7 Contents Contributors to this textbook v Overview 2 Systems of Linear Equations: Algebra 9 2. Systems of Linear Equations Row Reduction Parametric Form Systems of Linear Equations: Geometry Vectors Vector Equations and Spans Matrix Equations Solution Sets Linear Independence Subspaces Basis and Dimension Bases as Coordinate Systems The Rank Theorem and the Basis Theorem Linear Transformations and Matrix Algebra Matrix Transformations One-to-one and Onto Transformations Linear Transformations Matrix Multiplication Matrix Inverses Determinants Determinants: Definition Cofactor Expansions Determinants and Volumes Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors The Characteristic Polynomial Similarity vii

8 viii CONTENTS 6.4 Diagonalization Complex Eigenvalues Stochastic Matrices Orthogonality Dot Products and Orthogonality Orthogonal Complements Orthogonal Projection Orthogonal Sets The Method of Least Squares A Complex Numbers 393 B Notation 397 C Hints and Solutions to Selected Exercises 399 D GNU Free Documentation License 40 Index 4

9 Chapter Overview The Subject of This Textbook Before starting with the content of the text, we first ask the basic question: what is linear algebra? Linear: having to do with lines, planes, etc. Algebra: solving equations involving unknowns. The name of the textbook highlights an important theme: the synthesis between algebra and geometry. It will be very important to us to understand systems of linear equations both algebraically (writing equations for their solutions) and geometrically (drawing pictures and visualizing). Remark. The term algebra was coined by the 9th century mathematician Abu Ja far Muhammad ibn Muso al-khwarizmi. It comes from the Arabic word al-jebr, meaning reunion of broken parts. At the simplest level, solving a system of linear equations is not very hard. You probably learned in high school how to solve a system like x + 3 y z = 4 2x y + 3z = 7 y 4z = 3. However, in real life one usually has to be more clever. Engineers need to solve many, many equations in many, many variables. Here is a tiny example: 3x + 4x 2 + 0x 3 + 9x 4 2x 5 3x 6 = 4 7x + 2x 2 3x 3 7x 4 + 2x 5 + 8x 6 = 2567 x + 9x x 3 + x 4 + 4x x 6 = 26 2 x + 4x 2 + 0x 3 + x 4 + 2x 5 + x 6 = 5. Often it is enough to know some information about the set of solutions, without having to solve the equations in the first place. For instance, does there exist a solution? What does the solution set look like geometrically? Is there still a solution if we change the 26 to a 27?

10 2 CHAPTER. OVERVIEW Sometimes the coefficients also contain parameters, like the eigenvalue equation (7 λ)x + y + 3z = 0 3x + (2 λ)y 3z = 0 3x 2 y + ( λ)z = 0. In data modeling, a system of equations generally does not actually have a solution. In that case, what is the best approximate solution? Accordingly, this text is organized into three main sections.. Solve the matrix equation Ax = b (chapters 2 4). Solve systems of linear equations using matrices, row reduction, and inverses. Analyze systems of linear equations geometrically using the geometry of solution sets and linear transformations. 2. Solve the matrix equation Ax = λx (chapters 5 6). Solve eigenvalue problems using the characteristic polynomial. Understand the geometry of matrices using similarity, eigenvalues, diagonalization, and complex numbers. 3. Approximately solve the matrix equation Ax = b (chapter 7). Find best-fit solutions to systems of linear equations that have no actual solution using least-squares approximations. Study the geometry of closest vectors and orthogonal projections. This text is roughly half computational and half conceptual in nature. The main goal is to present a library of linear algebra tools, and more importantly, to teach a conceptual framework for understanding which tools should be applied in a given context. If Matlab can find the answer faster than you can, then your question is just an algorithm: this is not real problem solving. The subtle part of the subject lies in understanding what computation to ask the computer to do for you it is far less important to know how to perform computations that a computer can do better than you anyway.

11 Uses of Linear Algebra in Engineering The vast majority of undergraduates at Georgia Tech have to take a course in linear algebra. There is a reason for this: Most engineering problems, no matter how complicated, can be reduced to linear algebra: Ax = b or Ax = λx or Ax b. 3 Here we present some sample problems in science and engineering that require linear algebra to solve. Example (Civil Engineering). The following diagram represents traffic flow around the town square. The streets are all one way, and the numbers and arrows indicate the number of cars per hour flowing along each street, as measured by sensors underneath the roads. Traffic flow (cars/hr) x w y 75 5 z There are no sensors underneath some of the streets, so we do not know how much traffic is flowing around the square itself. What are the values of x, y, z, w? Since the number of cars entering each intersection has to equal the number of cars leaving that intersection, we obtain a system of linear equations: w + 20 = x x + 20 = y + 70 y = z z + 5 = w + 75.

12 4 CHAPTER. OVERVIEW Example (Chemical Engineering). A certain chemical reaction (burning) takes ethane and oxygen, and produces carbon dioxide and water: x C 2 H 6 + y O 2 z CO 2 + w H 2 O What ratio of the molecules is needed to sustain the reaction? The following three equations come from the fact that the number of atoms of carbon, hydrogen, and oxygen on the left side has to equal the number of atoms on the right, respectively: 2x = z 6x = 2w 2 y = 2z + w. Example (Biology). In a population of rabbits,. half of the newborn rabbits survive their first year; 2. of those, half survive their second year; 3. the maximum life span is three years; 4. rabbits produce 0, 6, 8 baby rabbits in their first, second, and third years, respectively. If you know the rabbit population in 206 (in terms of the number of first, second, and third year rabbits), then what is the population in 207? The rules for reproduction lead to the following system of equations, where x, y, z represent the number of newborn, first-year, and second-year rabbits, respectively: 6 y z 206 = x x 206 = y y 206 = z 207. A common question is: what is the asymptotic behavior of this system? What will the rabbit population look like in 00 years? This turns out to be an eigenvalue problem. Use this link to view the online demo Left: the population of rabbits in a given year. Right: the proportions of rabbits in that year. Choose any values you like for the starting population, and click Advance year several times. What do you notice about the long-term behavior of the ratios? This phenomenon turns out to be due to eigenvectors.

13 5 Example (Astronomy). An asteroid has been observed at the following locations: (0, 2), (2, ), (, ), (, 2), ( 3, ), (, ). Its orbit around the sun is elliptical; it is described by an equation of the form x 2 + B y 2 + C x y + Dx + E y + F = 0. What is the most likely orbit of the asteroid, given that there was some significant error in measuring its position? Substituting the data points into the above equation yields the system (0) 2 + B(2) 2 + C(0)(2) + D(0) + E(2) + F = 0 (2) 2 + B() 2 + C(2)() + D(2) + E() + F = 0 () 2 + B( ) 2 + C()( ) + D() + E( ) + F = 0 ( ) 2 + B( 2) 2 + C( )( 2) + D( ) + E( 2) + F = 0 ( 3) 2 + B() 2 + C( 3)() + D( 3) + E() + F = 0 ( ) 2 + B( ) 2 + C( )( ) + D( ) + E( ) + F = 0. There is no actual solution to this system due to measurement error, but here is the best-fitting ellipse: (0, 2) ( 3, ) (, ) (2, ) (, ) (, 2) 266x y 2 78x y + 402x 23 y 374 = 0 Example (Computer Science). Each web page has some measure of importance, which it shares via outgoing links to other pages. This leads to zillions of equations in zillions of variables. Larry Page and Sergei Brin realized that this is a linear algebra problem at its core, and used the insight to found Google. We will discuss this example in detail in Section 6.6.

14 6 CHAPTER. OVERVIEW How to Use This Textbook There are a number of different categories of ideas that are contained in most sections. They are listed at the top of the section, under Objectives, for easy review. We classify them as follows. Recipes: these are algorithms that are generally straightforward (if sometimes tedious), and are usually done by computer in real life. They are nonetheless important to learn and to practice. Vocabulary words: forming a conceptual understanding of the subject of linear algebra means being able to communicate much more precisely than in ordinary speech. The vocabulary words have precise definitions, which must be learned and used correctly. Essential vocabulary words: these vocabulary words are essential in that they form the essence of the subject of linear algebra. For instance, if you do not know the definition of an eigenvector, then by definition you cannot claim to understand linear algebra. Theorems: these describe in a precise way how the objects of interest relate to each other. Knowing which recipe to use in a given situation generally means recognizing which vocabulary words to use to describe the situation, and understanding which theorems apply to that problem. Pictures: visualizing the geometry underlying the algebra means interpreting and drawing pictures of the objects involved. The pictures are meant to be a core part of the material in the text: they are not just a pretty add-on. This textbook is exclusively targeted at Math 553 at Georgia Tech. As such, it contains exactly the material that is taught in that class; no more, and no less: students in Math 553 are responsible for understanding all visible content. In the online version some extra material (most examples and proofs, for instance) is hidden, in that one needs to click on a link to reveal it, like this: Hidden Content. Hidden content is meant to enrich your understanding of the topic, but is not an official part of Math 553. That said, the text will be very hard to follow without understanding the examples, and studying the proofs is an excellent way to learn the conceptual part of the material. (Not applicable to the PDF version.) Finally, we remark that there are over 40 interactive demos contained in the text, which were created to illustrate the geometry of the topic. Click the view in a new window link, and play around with them! You will need a modern browser. Internet Explorer is not a modern browser; try Safari, Chrome, or Firefox. Here is a demo from Section 7.5: Use this link to view the online demo Click and drag the points on the grid on the right.

15 Feedback Every page of the online version has a link on the bottom for providing feedback. This will take you to the GitHub Issues page for this book. It requires a Georgia Tech login to access. 7

16 8 CHAPTER. OVERVIEW

17 Chapter 2 Systems of Linear Equations: Algebra Primary Goal. Solve a system of linear equations algebraically in parametric form. This chapter is devoted to the algebraic study of systems of linear equations and their solutions. We will learn a systematic way of solving equations of the form 3x + 4x 2 + 0x 3 + 9x 4 2x 5 3x 6 = 4 7x + 2x 2 3x 3 7x 4 + 2x 5 + 8x 6 = 2567 x + 9x x 3 + x 4 + 4x x 6 = 26 2 x + 4x 2 + 0x 3 + x 4 + 2x 5 + x 6 = 5. In Section 2., we will introduce systems of linear equations, the class of equations whose study forms the subject of linear algebra. In Section 2.2, will present a procedure, called row reduction, for finding all solutions of a system of linear equations. In Section 2.3, you will see hnow to express all solutions of a system of linear equations in a unique way using the parametric form of the general solution. 2. Systems of Linear Equations Objectives. Understand the definition of R n, and how to use R n to label points. 2. Learn to replace a system of linear equations by an augmented matrix. 3. Learn how the elimination method corresponds to performing row operations on an augmented matrix. 4. Vocabulary words: consistent, inconsistent, solution set, row operation, row equivalence, matrix, augmented matrix. 9

18 0 CHAPTER 2. SYSTEMS OF LINEAR EQUATIONS: ALGEBRA 2.. Preliminaries: Line, Plane, Space, Etc. We use R to denote the set of all real numbers, i.e., the number line. This contains numbers like 0, 3 2, π, 04,... Definition. Let n be a positive whole number. We define R n = all ordered n-tuples of real numbers (x, x 2, x 3,..., x n ). In other words, R n is just the set of all (ordered) lists of n real numbers. We will draw pictures of R n in a moment, but keep in mind that this is the definition. For example, (0, 3 2, π) is an element of R3. Example (The number line). When n =, we just get R back: R = R. Geometrically, this is the number line Definition (Lines). For our purposes, a line is a ray that is straight and infinite in both directions. The number line R = R is an example of a line, but a line is not the same as R. We will see later how to use the elements of R to label the points on a line. Example (The plane). When n = 2, we can think of R 2 as the x y-plane. We can do so because every point on the plane can be represented by an ordered pair of real numbers, namely, its x- and y-coordinates.

19 2.. SYSTEMS OF LINEAR EQUATIONS (, 2) (0, 3) Note that we use the elements of R 2 to label the points of the plane, but R 2 is not defined to be the plane. Definition (Planes). A plane is a flat sheet that is infinite in all directions. z x y The x y-plane R 2 is an example of a plane, but a plane is not the same as R 2. We will see later how to use the elements of R 2 to label the points on a plane. Example (3-Space). When n = 3, we can think of R 3 as the space we (appear to) live in. We can do so because every point in space can be represented by an ordered triple of real numebrs, namely, its x-, y-, and z-coordinates.

20 2 CHAPTER 2. SYSTEMS OF LINEAR EQUATIONS: ALGEBRA (,, 3) ( 2, 2, 2) Again, we can use the elements of R 3 to label points in space, but R 3 is not defined to be space! Example (Color Space). All colors you can see can be described by three quantities: the amount of red, green, and blue light in that color. (Humans are trichromatic.) So we can also think of R 3 as the space of all colors. blue green red Again, we can use the elements of R 3 to label the colors, but R 3 is not defined to be the set of all visible colors! Example. In Chapter, we could have used R 4 to label the amount of traffic (x, y, z, w) passing through four streets.

21 2.. SYSTEMS OF LINEAR EQUATIONS 3 x w y z So what is R 4? or R 5? or R n? These are harder to visualize, so you have to go back to the definition: R n is the set of all ordered n-tuples of real numbers (x, x 2, x 3,..., x n ). They are still geometric spaces, in the sense that our intuition for R 2 and R 3 often extends to R n. We will make definitions and state theorems that apply to any R n, but we will only draw pictures for R 2 and R Solving Systems of Equations During the first half of this textbook, we will be primarily concerned with understanding the solutions to linear equations. Definition. An equation in the unknowns x, y, z,... is called linear if both sides of the equation are a sum of (constant) multiples of x, y, z,..., plus an optional constant. For instance, are linear equations, but 3x + 4 y = 2z x z = 00 3x + yz = 3 sin(x) cos(y) = 2 are not. We will usually move the unknowns to the left side of the equation, and move the constants to the right. A system of linear equations is a collection of several linear equations, like x + 2 y + 3z = 6 2x 3 y + 2z = 4 3x + y z = 2. (2..)

22 4 CHAPTER 2. SYSTEMS OF LINEAR EQUATIONS: ALGEBRA Definition (Solution sets). A solution of a system of equations is a list of numbers x, y, z,... that make all of the equations true simultaneously. The solution set of a system of equations is the collection of all solutions. Solving the system means describing the solution set parametrically. A solution of a system of equations in n variables is a list of n numbers: it is an element of R n. For example, (x, y, z) = (, 2, 3) is a solution of (2..); this is an element of R 3. The elimination method We will solve systems of linear equations using the elimination method. In other words, we will combine the equations in various ways to try to eliminate as many variables as possible from each equation. There are three valid operations we can perform on our system of equations: Scaling: we can multiply both sides of an equation by a nonzero number. x + 2 y + 3z = 6 2x 3 y + 2z = 4 3x + y z = 2 multiply st by 3 3x 6 y 9z = 8 2x 3 y + 2z = 4 3x + y z = 2 Replacement: we can add a multiple of one equation to another, replacing the second equation with the result. x + 2 y + 3z = 6 2x 3 y + 2z = 4 3x + y z = 2 2nd = 2nd 2 st x + 2 y + 3z = 6 7 y 4z = 2 3x + y z = 2 Swap: we can swap two equations. x + 2 y + 3z = 6 2x 3 y + 2z = 4 3x + y z = 2 3rd st 3x + y z = 2 2x 3 y + 2z = 4 x + 2 y + 3z = 6 Example. Solve (2..) using the elimination method.

23 2.. SYSTEMS OF LINEAR EQUATIONS 5 Solution. x + 2 y + 3z = 6 2x 3 y + 2z = 4 3x + y z = 2 2nd = 2nd 2 st 3rd = 3rd 3 st 2nd 3rd divide 2nd by 5 3rd = 3rd+7 2nd x + 2 y + 3z = 6 7 y 4z = 2 3x + y z = 2 x + 2y + 3z = 6 7y 4z = 2 5y 0z = 20 x + 2y + 3z = 6 5y 0z = 20 7y 4z = 2 x + 2 y + 3z = 6 y + 2z = 4 7 y 4z = 2 x + 2 y + 3z = 6 y + 2z = 4 0z = 30 At this point we ve eliminated both x and y from the third equation, and we can solve 0z = 30 to get z = 3. Substituting for z in the second equation gives y = 4, or y = 2. Substituting for y and z in the first equation gives x + 2 ( 2) = 6, or x = 3. Thus the only solution is (x, y, z) = (, 2, 3). We can check that our solution is correct by substituting (x, y, z) = (, 2, 3) into the original equation: x + 2 y + 3z = 6 2x 3 y + 2z = 4 3x + y z = 2 substitute + 2 ( 2) = ( 2) = ( 2) 3 = 2. Augmented Matrices and Row Operations Solving equations by elimination requires writing the variables x, y, z and the equals sign = over and over again, merely as placeholders: all that is changing in the equations is the coefficient numbers. We can make our life easier by extracting only the numbers, and putting them in a box: x + 2 y + 3z = becomes 2x 3 y + 2z = x + y z = This is called an augmented matrix. The word augmented refers to the vertical line, which we draw to remind ourselves where the equals sign belongs; a matrix is a grid of numbers without the vertical line. In this notation, our three valid ways of manipulating our equations become row operations:

24 6 CHAPTER 2. SYSTEMS OF LINEAR EQUATIONS: ALGEBRA Scaling: multiply all entries in a row by a nonzero number R =R Here the notation R simply means the first row, and likewise for R 2, R 3, etc. Replacement: add a multiple of one row to another, replacing the second row with the result R 2=R 2 2 R Swap: interchange two rows R R Example. Solve (2..) using row operations. Solution. We start by forming an augmented matrix: x + 2 y + 3z = 6 2x 3 y + 2z = 4 3x + y z = 2 becomes Eliminating a variable from an equation means producing a zero to the left of the line in an augmented matrix. First we produce zeros in the first column (i.e. we eliminate x) by subtracting multiples of the first row R 2=R 2 2R 3 2 R 3 =R 3 3R This was made much easier by the fact that the top-left entry is equal to, so we can simply multiply the first row by the number below and subtract. In order to eliminate y in the same way, we would like to produce a in the second column.

25 2.. SYSTEMS OF LINEAR EQUATIONS 7 We could divide the second row by 7, but this would produce fractions; instead, let s divide the third by R 3=R R 2 R R 2 R R 3 =R We swapped the second and third row just to keep things orderly. Now we translate this augmented matrix back into a system of equations: x + 2 y + 3z = becomes y + 2z = z = 3 Hence z = 3; back-substituting as in this example gives (x, y, z) = (, 2, 3). The process of doing row operations to a matrix does not change the solution set of the corresponding linear equations! Indeed, the whole point of doing these operations is to solve the equations using the elimination method. Definition. Two matrices are called row equivalent if one can be obtained from the other by doing some number of row operations. So the linear equations of row-equivalent matrices have the same solution set. Example (An Inconsistent System). Solve the following system of equations using row operations: x + y = 2 3x + 4 y = 5 4x + 5 y = 9 Solution. First we put our system of equations into an augmented matrix. x + y = 2 2 augmented matrix 3x + 4 y = x + 5 y =

26 8 CHAPTER 2. SYSTEMS OF LINEAR EQUATIONS: ALGEBRA We clear the entries below the top-left using row replacement R 2=R 2 3R R 3 =R 3 4R Now we clear the second entry from the last row. 2 0 R 3=R 3 R 2 0 This translates back into the system of equations x + y = 2 y = 0 = Our original system has the same solution set as this system. But this system has no solutions: there are no values of x, y making the third equation true! We conclude that our original equation has no solutions either. Definition. A system of equations is called inconsistent if it has no solutions. It is called consistent otherwise. 2.2 Row Reduction Objectives. Understand when a matrix is in (reduced) row echelon form. 2. Learn which row reduced matrices come from inconsistent linear systems. 3. Recipe: the row reduction algorithm. 4. Vocabulary words: pivot, (reduced) row echelon form.

27 2.2. ROW REDUCTION Echelon Forms In Section 2. we saw how to translate a system of linear equations into an augmented matrix. We want to find an algorithm for solving such an augmented matrix. First we must decide what it means for an augmented matrix to be solved. Definition. A matrix is in row echelon form if:. All zero rows are at the bottom. 2. Each leading nonzero entry of a row is to the right of the leading entry of the row above. 3. Below a leading entry of a row, all entries are zero. Here is a picture of a matrix in row echelon form: 0 = any number = any nonzero number Definition. A pivot is the first nonzero entry of a row of a matrix in row echelon form. Definition. A matrix is in reduced row echelon form if it is in row echelon form, and in addition: 5. The pivot in each nonzero row is equal to. 6. Each pivot is the only nonzero entry in its column. Here is a picture of a matrix in reduced row echelon form: = any number = pivot Example. The following matrices are in reduced row echelon form: The following matrix is in row echelon form but not reduced row echelon form: The following matrix is not in echelon form:

28 20 CHAPTER 2. SYSTEMS OF LINEAR EQUATIONS: ALGEBRA When deciding if an augmented matrix is in (reduced) row echelon form, there is nothing special about the augmented column(s). Just ignore the vertical line. If an augmented matrix is in reduced row echelon form, the corresponding linear system is viewed as solved. We will see below why this is the case, and we will show that any matrix can be put into reduced row echelon form using only row operations The Row Reduction Algorithm Theorem. Every matrix is row equivalent to one and only one matrix in reduced row echelon form. We will give an algorithm, called row reduction or Gaussian elimination, which demonstrates that every matrix is row equivalent to at least one matrix in reduced row echelon form. The uniqueness statement is interesting it means that, no matter how you row reduce, you always get the same matrix in reduced row echelon form. This assumes, of course, that you only do the three legal row operations, and you don t make any arithmetic errors. We will not prove uniqueness, but maybe you can! Algorithm (Row Reduction). Step a: Swap the st row with a lower one so a leftmost nonzero entry is in the st row (if necessary). Step b: Scale the st row so that its leading entry is equal to. Step c: Use row replacement so all entries below this are 0. Step 2a: Swap the 2nd row with a lower one so that the leftmost nonzero entry is in the 2nd row. Step 2b: Scale the 2nd row so that its leading entry is equal to. Step 2c: Use row replacement so all entries below this are 0. Step 3a: Swap the 3rd row with a lower one so that the leftmost nonzero entry is in the 3rd row. etc.

29 2.2. ROW REDUCTION 2 Last Step: Use row replacement to clear all entries above the pivots, starting with the last pivot. Example. Row reduce this matrix: Solution.

30 22 CHAPTER 2. SYSTEMS OF LINEAR EQUATIONS: ALGEBRA Step a: Row swap to make this nonzero. Optional: swap rows 2 and 3 to make Step 2b easier next. Note how Step 2b doesn t create fractions. R R 2 R = R 2 R 3 = R 3 3R R 2 R 3 R 2 = R 2 5 R 3 = R 3 + 7R 2 R 3 = R 3 0 R 2 = R 2 2R Step b: Scale to make this Step c: Subtract a multiple of the first row to clear this Step 2a: This is already nonzero. Step 2b: Scale to make this Step 2c: Add 7 times the second row to clear this Step 3a: This is already nonzero. Step 3b: Scale to make this Last step: add multiples of the third row to clear these R = R 3R 3 R = R 2R Last step: add 2 times the third row to clear this The reduced row echelon form of the matrix is 0 0 x = translates to y = z = 3.

31 2.2. ROW REDUCTION 23 The reduced row echelon form of the matrix tells us that the only solution is (x, y, z) = (, 2, 3). Use this link to view the online demo Animated slideshow of the row reduction in this example. Here is the row reduction algorithm, summarized in pictures. Get a here Clear down Get a here Clear down (maybe these are already zero) Get a here Clear down Matrix is in REF Clear up Clear up Matrix is in RREF Profit? Example. Solve the linear system using row reduction. 2x + 0y = 3x + 5y = 2

32 24 CHAPTER 2. SYSTEMS OF LINEAR EQUATIONS: ALGEBRA Solution R =R 2 R 2 =R 2 3R R 2 =R R =R + 2 R (Step b) (Step c) (Step 2b) (Step 2c) This row reduced matrix corresponds to the inconsistent system x + 5 y = 0 0 =. The Row Echelon Form of an Inconsistent System. An augmented matrix corresponds to an inconsistent system of equations if and only if the last column (i.e., the augmented column) is a pivot column. In other words, the row reduced matrix of an inconsistent system looks like this: Parametric Form Objectives. Learn to express the solution set of a system of linear equations in parametric form. 2. Understand the three possibilities for the number of solutions of a system of linear equations. 3. Recipe: parametric form. 4. Vocabulary word: free variable.

33 2.3. PARAMETRIC FORM Free Variables There is one possibility for the row reduced form of a matrix that we did not see in Section 2.2. Example (A System with a Free Variable). Solve the linear system 2x + y + 2z = x + 2 y + 9z = using row reduction. Solution R R 2 R 2 =R 2 2R R 2 =R 2 3 R =R 2R (Optional) (Step c) (Step 2b) (Step 2c) This row reduced matrix corresponds to the linear system x + 5z = y + 2z =. In what sense is the system solved? We rewrite as x = 5z y = 2z. For any value of z, there is exactly one value of x and y that make the equations true. But we are free to choose any value of z. We have found all solutions: it is the set of all values x, y, z, where x = 5z y = 2z z any real number. For instance, setting z = 0 gives the solution (x, y, z) = (,, 0), and setting z = gives the solution (x, y, z) = ( 4, 3, ). Use this link to view the online demo A picture of the solution set (the yellow line) of the linear system in this example. There is a unique solution for every value of z; move the slider to change z.

34 26 CHAPTER 2. SYSTEMS OF LINEAR EQUATIONS: ALGEBRA Definition. Consider a consistent system of equations in the variables x, x 2,..., x n. Let A be a row echelon form of the augmented matrix for this system. We say that x i is a free variable if its corresponding column in A does not contain a pivot. In the above example, the variable z was free because the reduced row echelon form matrix was In the matrix 0, 0 0 the free variables are x 2 and x 4. (The augmented column is not free because it does not correspond to a variable.) Recipe: Parametric form. The parametric form of the solution set of a consistent system of linear equations is obtained as follows.. Write the system as an augmented matrix. 2. Row reduce to reduced row echelon form. 3. Write the corresponding (solved) system of linear equations. 4. Move all free variables to the right hand side of the equations. Moving the free variables to the right hand side of the equations amounts to solving for the non-free variables (the ones that come from columns with pivots) in terms of the free variables. One can think of the free variables as being independent variables, and the non-free variables being dependent. Implicit Versus Parameterized Equations. The solution set of the system of linear equations 2x + y + 2z = x + 2 y + 9z = is a line in R 3, as we saw in this example. These equations are called the implicit equations for the line: the line is defined implicitly as the simultaneous solutions to those two equations. The parametric form (x, y, z) = ( 5z, 2z, z) z any real number is called a parameterized equation for the same line. It is an expression that produces all points of the line in terms of one parameter, z. One should think of a system of equations as being an implicit equation for its solution set, and of the parametric form as being the parameterized equation for the same set. The parameteric form is much more explicit: it gives a concrete recipe for producing all solutions.

35 2.3. PARAMETRIC FORM 27 You can chose any value for the free variables in a (consistent) linear system. Free variables come from the columns without pivots in a matrix in row echelon form. Example. Suppose that the reduced row echelon form of the matrix for a linear system in four variables x, x 2, x 3, x 4 is The free variables are x 2 and x 4 : they are the ones whose columns are not pivot columns. This translates into the system of equations x + 3x 4 = 2 x 3 + 4x 4 = parametric form x = 2 3x 4 x 3 = 4x 4. What happened to x 2? It is a free variable, but no other variable depends on it. The general solution to the system is (x, x 2, x 3, x 4 ) = (2 3x 4, x 2, 4x 4, x 4 ), for any values of x 2 and x 4. For instance, (2, 0,, 0) is a solution (with x 2 = x 4 = 0), and (5,, 3, ) is a solution (with x 2 =, x 4 = ). Example (A Parameterized Plane). The system of one linear equation x + y + z = comes from the matrix, which is already in reduced row echelon form. The free variables are y and z. The parametric form for the general solution is (x, y, z) = ( y z, y, z) for any values of y and z. This is the parametric equation for a plane in R 3. Use this link to view the online demo A plane described by two parameters y and z. Any point on the plane is obtained by substituting suitable values for y and z.

36 28 CHAPTER 2. SYSTEMS OF LINEAR EQUATIONS: ALGEBRA Number of Solutions There are three possibilities for the reduced row echelon form of the augmented matrix of a linear system.. The last column is a pivot column. In this case, the system is inconsistent. There are zero solutions, i.e., the solution set is empty. For example, the matrix comes from a linear system with no solutions. 2. Every column except the last column is a pivot column. In this case, the system has a unique solution. For example, the matrix 0 0 a 0 0 b 0 0 c tells us that the unique solution is (x, y, z) = (a, b, c). 3. The last column is not a pivot column, and some other column is not a pivot column either. In this case, the system has infinitely many solutions, corresponding to the infinitely many possible values of the free variable(s). For example, in the system corresponding to the matrix 2 0 3, any values for x 2 and x 4 yield a solution to the system of equations.

37 Chapter 3 Systems of Linear Equations: Geometry Primary Goals.. Understand what the solution set of Ax = b looks like. 2. Understand the set of b such that Ax = b is consistent. This chapter is devoted to the geometric study of two objects:. the solution set of a system of linear equations, and 2. the set of all constants that makes a particular system consistent. These objects are related in a beautiful way by the rank theorem in Section 3.9. We will develop a large amount of vocabulary that we will use to describe the above objects: vectors (Section 3.), spans (Section 3.2), linear independence (Section 3.5), subspaces (Section 3.6), dimension (Section 3.7), coordinate systems (Section 3.8), etc. We will use these concepts to give a precise geometric description of the solution set of any system of equations (Section 3.4). We will also learn how to express systems of equations more simply using matrix equations (Section 3.3). 3. Vectors Objectives. Learn how to add and scalar-multiply vectors in R n, both algebraically and geometrically. 2. Understand linear combinations geometrically. 3. Pictures: vector addition, vector subtraction, linear combinations. 29

38 30 CHAPTER 3. SYSTEMS OF LINEAR EQUATIONS: GEOMETRY 4. Vocabulary words: vector, linear combination. 3.. Vectors in R n We have been drawing elements of R n as points in the line, plane, space, etc. We can also draw them as arrows. Points and Vectors. A point is an element of R n, drawn as a point (a dot). the point (, 3) A vector is an element of R n, drawn as an arrow. the vector 3 The difference is purely psychological: points and vectors are both just lists of numbers. Interactive: A vector in R 3, by coordinates. Use this link to view the online demo A vector in R 3, and its coordinates. Drag the arrow head and tail.

39 3.. VECTORS 3 When we think of an element of R n as a vector, we will usually write it vertically, like a matrix with one column: v =. 3 We will also write 0 for the zero vector. Why make the distinction between points and vectors? A vector need not start at the origin: it can be located anywhere! In other words, an arrow is determined by its length and its direction, not by its location. For instance, these arrows all represent the vector. 2 Unless otherwise specified, we will assume that all vectors start at the origin. Remark. Vectors makes sense in the real world: many physical quantities, such as velocity, are represented as vectors. But it makes more sense to think of the velocity of a car as being located at the car. Remark. Some authors use boldface letters to represent vectors, as in v, or use arrows, as in v. As it is usually clear from context if a letter represents a vector, we do not decorate vectors in this way. Note. Another way to think about a vector is as a difference between two points, or the arrow from one point to another. For instance, 2 is the arrow from (, ) to (2, 3). (, ) 2 (2, 3)

40 32 CHAPTER 3. SYSTEMS OF LINEAR EQUATIONS: GEOMETRY 3..2 Vector Algebra and Geometry Here we learn how to add vectors together and how to multiply vectors by numbers, both algebraically and geometrically. Vector addition and scalar multiplication. We can add two vectors together: a x a + x b + y = b + y. c z c + z We can multiply, or scale, a vector by a real number c: x c x c y = c y. z c z We call c a scalar to distinguish it from a vector. If v is a vector and c is a scalar, then cv is called a scalar multiple of v. Addition and scalar multiplication work in the same way for vectors of length n. Example = 7 and 2 2 = The Parallelogram Law for Vector Addition Geometrically, the sum of two vectors v, w is obtained as follows: place the tail of w at the head of v. Then v + w is the vector whose tail is the tail of v and whose head is the head of w. Doing this both ways creates a parallelogram. For example, = Why? The width of v + w is the sum of the widths, and likewise with the heights.

41 3.. VECTORS 33 w 5 = = v w v + w v 5 = + 4 = 4 + Interactive: The parallelogram law for vector addition. Use this link to view the online demo The parallelogram law for vector addition. Click and drag the heads of and w. Vector Subtraction Geometrically, the difference of two vectors v, w is obtained as follows: place the tail of v and w at the same point. Then v w is the vector from the head of w to the head of v. For example, 4 3 = Why? If you add v w to w, you get v. v w v w Interactive: Vector subtraction. Use this link to view the online demo Vector subtraction. Click and drag the heads of v and w.

42 34 CHAPTER 3. SYSTEMS OF LINEAR EQUATIONS: GEOMETRY Scalar Multiplication A scalar multiple of a vector v has the same (or opposite) direction, but a different length. For instance, 2v is the vector in the direction of v but twice as long, and 2 v is the vector in the opposite direction of v, but half as long. Note that the set of all scalar multiples of a (nonzero) vector v is a line. Some multiples of v. All multiples of v. 2v v 2 v 0v Interactive: Scalar multiplication. Use this link to view the online demo Scalar multiplication. Drag the slider to change the scalar Linear Combinations We can add and scalar-multiply vectors in the same equation. Definition. Let c, c 2,..., c p be scalars, and let v, v 2,..., v p be vectors in R n. The vector in R n c v + c 2 v c p v p is called a linear combination of the vectors v, v 2,..., v p, with weights or coefficients c, c 2,..., c p. Geometrically, a linear combination is obtained by stretching / shrinking the vectors v, v 2,..., v p according to the coefficients, then adding them together using the parallelogram law. Example. Let v = 2 and v2 = 0. Here are some linear combinations of v and v 2, drawn as points.

43 3.. VECTORS 35 v + v 2 v v v 2 2v + 0v 2 v 2 2v 2 v The locations of these points are found using the parallelogram law for vector addition. Any vector on the plane is a linear combination of v and v 2, with suitable coefficients. Use this link to view the online demo Linear combinations of two vectors in R 2 : move the sliders to change the coefficients of v and v 2. Note that any vector on the plane can be obtained as a linear combination of v, v 2 with suitable coefficients. Interactive: Linear combinations of three vectors. Use this link to view the online demo Linear combinations of three vectors: move the sliders to change the coefficients of v, v 2, v 3. Note how the parallelogram law for addition of three vectors is more of a parallepiped law. Example (Linear Combinations of a Single Vector). A linear combination of a single vector v = 2 is just a scalar multiple of v. So some examples include v = 2, 3 2 v = 3/2 3, /2 2 v =,... The set of all linear combinations is the line through v. (Unless v = 0, in which case any scalar multiple of v is again 0.)

44 36 CHAPTER 3. SYSTEMS OF LINEAR EQUATIONS: GEOMETRY v Example (Linear Combinations of Collinear Vectors). The set of all linear combinations of the vectors 2 v = and v 2 2 = is the line containing both vectors. v v 2 The difference between this and a previous example is that both vectors lie on the same line. Hence any scalar multiples of v, v 2 lie on that line, as does their sum. Interactive: Linear combinations of two collinear vectors. Use this link to view the online demo Linear combinations of two collinear vectors in R 2. Move the sliders to change the coefficients of v, v 2. Note that there is no way to escape the line.

45 3.2. VECTOR EQUATIONS AND SPANS Vector Equations and Spans Objectives. Understand the equivalence between a system of linear equations and a vector equation. 2. Learn the definition of Span{x, x 2,..., x p }, and how to draw pictures of spans. 3. Recipe: solve a vector equation using augmented matrices / decide if a vector is in a span. 4. Pictures: an inconsistent system of equations, a consistent system of equations, spans in R 2 and R Vocabulary word: vector equation. 6. Essential vocabulary word: span Vector Equations An equation involving vectors with n coordinates is the same as n equations involving only numbers. For example, the equation 8 x 2 + y 2 = 6 (3.2.) 6 3 simplifies to x y 8 2x + 2 y = 6 6x y 3 or x y 8 2x 2 y = 6. 6x y 3 For two vectors to be equal, all of their coordinates must be equal, so this is just the system of linear equations x y = 8 2x 2 y = 6 6x y = 3. (3.2.2) Definition. A vector equation is an equation involving a linear combination of vectors with possibly unknown coefficients.

46 38 CHAPTER 3. SYSTEMS OF LINEAR EQUATIONS: GEOMETRY 8 Example. Is 6 a linear combination of 2 and 2? 3 6 Solution. This means: can we solve the equation 8 x 2 + y 2 = in the unknowns x, y? This vector equation translates into the system of linear equations x y = 8 2x 2 y = 6 6x y = 3. We form an augmented matrix and row reduce: RREF Thus the equation is consistent, and the solution is x = and y = 9. We 8 conclude that 6 is indeed a linear combination of 2 and 2, with coefficients 3 6 and 9: = Use this link to view the online demo A picture of the vector equation (3.2.). Try to solve the equation geometrically by moving the sliders. A Picture of a Consistent System. We saw in the above example that the system of equations (3.2.2) is consistent. Equivalently, this means that the vector equation (3.2.) has a solution. Therefore, the figure above is a picture of a consistent system of equations. Compare this figure below. In order to actually solve the vector equation 8 x 2 + y 2 = 6, 6 3

47 3.2. VECTOR EQUATIONS AND SPANS 39 one has to solve the system of linear equations x y = 8 2x 2 y = 6 6x y = 3. This means forming the augmented matrix and row reducing. Note that the columns of the augmented matrix are the vectors from the original vector equation, so it is not actually necessary to write the system of equations: one can go directly from the vector equation to the augmented matrix by smooshing the vectors together. Recipe: Solving a vector equation. In general, the vector equation x v + x 2 v x p v p = b where v, v 2,..., v p, b are vectors in R n and x, x 2,..., x p are unknown scalars, has the same solution set as the linear system with augmented matrix v v 2 v p b whose columns are the v i s and the b s. Now we have (at least) two equivalent ways of thinking about systems of equations:. Augmented matrices. 2. Linear combinations of vectors (vector equations). The second is more geometric in nature: it lends itself to drawing pictures Spans It will be important to know what are all linear combinations of a set of vectors v, v 2,..., v p in R n. In other words, we would like to understand the set of all vectors b in R n such that the vector equation (in the unknowns x, x 2,..., x p ) has a solution (i.e. is consistent). x v + x 2 v x p v p = b

48 40 CHAPTER 3. SYSTEMS OF LINEAR EQUATIONS: GEOMETRY Essential Definition. Let v, v 2,..., v p be vectors in R n. The span of v, v 2,..., v p is the collection of all linear combinations of v, v 2,..., v p, and is denoted Span{v, v 2,..., v p }. In symbols: Span{v, v 2,..., v p } = x v + x 2 v x p v p x, x 2,..., x p in R We also say that Span{v, v 2,..., v p } is the subset spanned by or generated by the vectors v, v 2,..., v p. The above definition is the first of several essential definitions that we will see in this textbook. They are essential in that they form the essence of the subject of linear algebra: learning linear algebra means (in part) learning these definitions. All of the definitions are important, but it is essential that you learn and understand the definitions marked as such. Set Builder Notation. The notation x v + x 2 v x p v p x, x 2,..., x p in R reads as: the set of all things of the form x v + x 2 v x p v p such that x, x 2,..., x p are in R. The vertical line is such that ; everything to the left of it is the set of all things of this form, and everything to the right is the condition that those things must satisfy to be in the set. Specifying a set in this way is called set builder notation. All mathematical notation is only shorthand: any sequence of symbols must translate into a usual sentence. Three characterizations of consistency. Now we have three equivalent ways of making the same statement:. A vector b is in the span of v, v 2,..., v p. 2. The vector equation has a solution. x v + x 2 v x p v p = b 3. The linear system with augmented matrix v v 2 v p b is consistent. Equivalent means that, for any given list of vectors v, v 2,..., v p, b, either all three statements are true, or all three statements are false.

49 3.2. VECTOR EQUATIONS AND SPANS 4 Use this link to view the online demo This is a picture of an inconsistent linear system: the vector w on the right-hand side of the equation x v + x 2 v 2 = w is not in the span of v, v 2. Convince yourself of this by trying to solve the equation x v + x 2 v 2 = w by moving the sliders, and by row reduction. Compare this figure. Pictures of Spans Drawing a picture of Span{v, v 2,..., v p } is the same as drawing a picture of all linear combinations of v, v 2,..., v p. Span{v} Span{v, w} v v w Span{v, w} v w Pictures of spans in R 2. Interactive: Span of two vectors in R 2. Use this link to view the online demo Interactive picture of a span of two vectors in R 2. Check Show x.v + y.w and move the sliders to see how every point in the red region is in fact a linear combination of the two vectors.

50 42 CHAPTER 3. SYSTEMS OF LINEAR EQUATIONS: GEOMETRY Span{v} Span{v, w} v v w Span{u, v, w} Span{u, v, w} u w v w v u Pictures of spans in R 3. The span of two noncollinear vectors is the plane containing the origin and the heads of the vectors. Note that three coplanar (but not collinear) vectors span a plane and not a 3-space, just as two collinear vectors span a line and not a plane. Interactive: Span of two vectors in R 3. Use this link to view the online demo Interactive picture of a span of two vectors in R 3. Check Show x.v + y.w and move the sliders to see how every point in the red region is in fact a linear combination of the two vectors. Interactive: Span of three vectors in R 3. Use this link to view the online demo Interactive picture of a span of three vectors in R 3. Check Show x.v + y.w + z.u and move the sliders to see how every point in the red region is in fact a linear combination of the three vectors.

51 3.3. MATRIX EQUATIONS Matrix Equations Objectives. Understand the equivalence between a system of linear equations, an augmented matrix, a vector equation, and a matrix equation. 2. Characterize the vectors b such that Ax = b is consistent, in terms of the span of the columns of A. 3. Characterize matrices A such that Ax = b is consistent for all vectors b. 4. Recipe: multiply a vector by a matrix (two ways). 5. Picture: the set of all vectors b such that Ax = b is consistent. 6. Vocabulary word: matrix equation The Matrix Equation Ax = b. In this section we introduce a very concise way of writing a system of linear equations: Ax = b. Here A is a matrix and x, b are vectors (generally of different sizes), so first we must explain how to multiply a matrix by a vector. When we say A is an m n matrix, we mean that A has m rows and n columns. Remark. In this book, we do not reserve the letters m and n for the numbers of rows and columns of a matrix. If we write A is an n m matrix, then n is the number of rows of A and m is the number of columns. Definition. Let A be an m n matrix with columns v, v 2,..., v n : A = v v 2 v n The product of A with a vector x in R n is the linear combination x x Ax = v v 2 v n 2. = x v + x 2 v x n v n. This is a vector in R m. x n