Coordinates and Resources. Linear Algebra Lecture # 1: Introduction (...and who cares about Linear Algebra anyway?) Lecture Notes

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1 Linear Algebra Lecture # 1: Introduction (...and who cares about Linear Algebra anyway?) Monson H. Hayes mhh3@gatech.edu Coordinates and Resources My office is in Room 1002, Building 305. My address is mhh3@gatech.edu. Course notes, homework and solutions, handouts, and other useful resources are available on the Linear Algebra Web Page, This page is accessible from my CAU web page where you may find other information about me. Textbook: Howard Anton and Chris Rorres, Elementary Linear Algebra with Applications, John Wiley and Sons, 10th Edition, 2011 This material is the property of the author and is for the sole and exclusive use of his students. It is not for publication, nor is it to be sold, reproduced, or generally distributed. M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 Structure of the Course 1 Lectures Goal: To introduce concepts in linear algebra, and motivate their use and importance through fun and interesting problems and applications. Note: I cannot cover everything in 2.5 hours a week. It is up to you to supplement lectures with reading and outside classwork. Important! 2 Assignments Purpose: To give you a chance to exercise your mind, and to solidify the concepts introduced to you in class. Structure: Some simple drill problems, some more challenging thought problems, and some linear algebra projects. Importance: Not important unless you want to learn the material and get a good grade. Indisputable Fact: You will not understand linear algebra if you only sit back and listen and do not work problems. 3 Exams: Midterm and a Final Lecture Notes Before each class, lecture slides will be posted on the course web page. It is highly recommended that you review/read the lecture slides before class so that you are better prepared. Sometimes the slides that are presented will differ from those that have been posted. Sometimes, I will leave parts of the slide blank and fill it in during class, and many times I will create a new slide, in real-time, during lecture to answer a question or add some additional explanations. It will be your responsibility to take notes M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42

2 Grading and Class Attendance Grading and Class Attendance Midterm 35% Final 40% Assignments 05% ++ Computer Assignments 10% Class Participation 10% Elements that will enter into your Class Participation Grade 1 Your participation in class in asking/answering questions, 2 Your out-of-class discussions with me, after class or during office hours. Midterm 35% Final 40% Assignments 05% ++ Computer Assignments 10% Class Participation 10% Some notes on the Assignments 1 These are for you and for your benefit. 2 If I believed that you would work hard and put in the time to complete these assignments, I would not collect them for grading. 3 Discussing the problems with each other is encouraged, 4 Copying each others assignments or submitting solutions/answers on the web is strictly prohibited (I am very good at determining when this has been done). M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 Phones and Texting So There is one very simple rule in this course, and it is based on the following premise, which I hope is true: You are coming to class to learn and to participate in classroom discussions 1 Turn your phones off and put them away. 2 I do not to hear a phone ring during class or see you texting away under the desk. My Goals I have four goals for this course 1 To teach you something. 2 To have you appreciate the importance of linear algebra. 3 To help you break an unfortunate pattern in today s world of looking for the answer on the internet. You have a brain, and I want to give you the confidence to use it. 4 To have some fun. By the end of the term I hope that you can say I really enjoyed this course. M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42

3 Objectives of the Course 1 To understand basic operations with vectors and matrices, 2 To become familiar with the concept of a system of linear equations, and to understand the number and geometry of the solutions to these equations and how to find them, 3 To understand the concept of a vector space, both algebraically and abstractly, Course Prerequisites There are NO prerequisites for this course except a course in calculus, and: 1 A willingness to work hard and to think independently. 2 A commitment to put a time, outside of class, reading and working problems. 3 A desire to learn and to not be afraid to ask questions. 4 To be able to formulate a variety of problems in terms of finding the solution to a system of linear equations and to be able to efficiently find the solution. M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 Some Important Things to Understand About This Course Math is Not a Spectator Sport You cannot learn mathematics by just going to class and watching the instructor lecture and work problems. You need to get involved in the learning process. If you aren t willing to be actively involved in the process of learning mathematics, both inside and outside of the class room, then you will have trouble passing any math class. Please read How to Study Math which is available from the Linear Algebra web page under References. Some Important Things to Understand About This Course Work to Understand the Principles You can pass a history class by simply memorizing a set of dates, names and events. You will find, however, that in order to pass a math class you will need to do more than just memorize a set of formulas. You need to understand how to USE the formulas and that is often far different from just memorizing them. You also need to understand WHAT formula to use and WHEN. You may have to modify the formula to fit the problem. M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42

4 Some Important Things to Understand About This Course Mathematics is Cumulative You must always remember that mathematics courses are cumulative. Almost everything you do in a math class will depend on subjects that you ve previously learned. This goes beyond just knowing the previous sections in your current class to needing to remember material from previous classes. How Do I Study/Learn Mathematics? There are many things that are important for you to do in order to succeed. A few of the more important things, and ones that are most often not done are: 1 Do the homework in a manner that is conducive to learning. 2 Use the Internet as a resource for learning, but never as a place to look for answers to problems. 3 Do not wait until the last minute to do your homework, read the book, study your notes, or get ready for an exam. 4 Ask questions, seek help from your professor (it is his job and responsibility to help you learn - that is what (s)he is paid to do), discuss the course with classmates. M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 So Who Cares About Linear Algebra Anyway? Some (Interesting) and Important Applications Linear Algebra, perhaps more than any other course you will ever take in mathematics, is probably the most important course you will ever take. Linear algebra can be fun, it can be visual, and it is very powerful. Very powerful and extraordinarily useful programs are available to bring the full power of linear algebra to solve a problem. Examples are MATLAB and Octave. If one were to look at the applications where linear algebra plays an important role, one will immediately realize how important this subject is. Please read the Handout Applications of Linear Algebra that is available in the Reference section of the course web page. Coding theory, Cryptography, Tomography, Vector graphics, Games, Quantum computing, Economics, Chemistry, Genetics, Image compression, Markov chains Linear Regression M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42

5 Course Topics Today s Assignment Systems of Linear Equations and Matrices Determinants Euclidean Vector Spaces General Vector Spaces Eigenvalues and Eigenvectors Inner Product Spaces Diagonalization and Quadratic Forms Linear Transformations Applications Begin reading your text. pages Print a copy of the first assignment, read it, and begin working on it. Get familiar with the course web page, its resources and materials. Get MATLAB (Student Edition) or Octave installed on your computer (see course web page). The Campus Bookstore has MATLAB for KRW 55,000. The package includes: 1 MATLAB 2 Simulink 3 Seven Toolboxes: Symbolic Math Toolbox, Control System Toolbox, Signal Processing Toolbox, Signal Processing Blockset, Statistics Toolbox, Optimization Toolbox, Image Processing Toolbox Read some or all of the papers provided to you on the course web page. M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 What is Linear Algebra? Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. It has extensive applications in engineering, physics, natural sciences, computer science, and the social sciences (particularly in economics). Nonlinear mathematical models can often be approximated by linear ones. The study of linear algebra and matrices has a long history. It first emerged from determinants, which were used to solve systems of linear equations. Determinants were used by Leibnitz in 1693, and subsequently by Cramer in 1750 for solving linear systems. Linear Equations A linear equation is a function of one or more variables that involves sums of products of a scalar (number) with a variable. Example: The relationship between temperature in degrees Centigrade and degrees Fahrenheit is a linear equation, C = 5 9 (F 32) Rearranging this equation into the form that will be useful when solving systems of linear equations, we have Here, F and C are the variables. Another Example F 9 5 C = 32 4x1 + 6x2 = 3 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42

6 Linear Equations in Two Variables Linear Equations in Two Variables Important Concept A linear equation in two variables defines a line in a two-dimensional space. C Example: Consider, again, the linear equation defining the relationship between degrees Centigrade and degrees Fahrenheit, F 9 5 C = 32 Plotting the values of F and C for which this equation is valid produces a line in a two-dimensional space (plane) F F 9 5 C = 32 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 Linear Equations in Three Variables Linear Equations in Three Variables Important Concept A linear equation in three variables defines a plane in a three-dimensional space. x 3 Example x1 + 2x2 x3 = 1 Plotting the values of x1,x2, and x3 that satisfy this equation defines a two-dimensional plane in a three-dimensional space. x 1 x 2 ax1 + bx2 + cx3 = d M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42

7 Understanding the Dimension of the Solution Space Consider the linear equation x1 = 2 Depending on the problem/application, this equation may be viewed in many different ways. 1 As a function of one variable, x1 = 2 is an equation that represents a point in a one-dimensional space. Understanding the Dimension of the Solution Space 2 As a function of two variables, x1 and x2, x1 + 0 x2 = 2 where the coefficient multiplying the variable x2 is equal to zero, this equation represents a line in a two-dimensional space. x x 1 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 Understanding the Dimension of the Solution Space 2 As a function of three variables, x1, x2, and x3, x1 + 0 x2 + 0 x3 = 2 where the coefficients multiplying the variables x1 and x2 are equal to zero, this equation represents a plane in a three-dimensional space. x 3 A General Linear Equation A general linear equation in n variables is given by a1x1 + a2x2 + + anxn = b or, using the summation symbol,, as n akxk = b k=1 This equation represents an n-1 dimensional hyperplane in an n dimensional space (difficult to draw, so it is important to have a conceptual understanding of what this equation represents). x 2 2 x 1 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42

8 Nonlinear Equations We have looked at examples of linear equations. Some examples of nonlinear equations are: Systems of Linear Equations A system of linear equations, or a Linear System is a set of one or more linear equations involving the same set of variables. 1 Example 4x1 + 2x2 = x1x2 x2 2 x1 = 7 x1 2 + x2 2 = 25 y = sin(x) 2 Another Example 3 Yet Another Example 3x1 + 2x2 = 0 x1 + 4x2 = 1 5x1 4x2 + 3x3 = 1 7x1 + 4x2 8x3 = 5 9x1 + 2x2 = 4 2x1 3x2 = 2 7x1 2x2 = 5 What is the important difference between these sets of equations? M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 Systems of Linear Equations and Solution Sets The solution of a linear system, if one exists, is a set of values for the variables x1, x2,..., xn that make each equation in the set true. The solution set may be a single point, a line, a plane, or a hyperplane in n-dimensional space. It is possible that no solution exists (the equations are inconsistent), as illustrated in the following example: x1 + x2 = 1 x1 + x2 = 2 Linear Equation and a Linear System Important Concepts 1 A linear equation in n variables defines an n-dimensional hyperplane in an n-dimensional space. 2 A linear system is a set of m linear equations in n variables, and the system defines a set of m hyperplanes in an n-dimensional space. A linear equation: a1x1 + a2x2 + + anxn = b A linear system of m linear equations in n variables (unknowns): a11x1 + a12x2 + + a1nxn = b1 a21x1 + a22x2 + + a2nxn = b2.. am1x1 + am2x2 + + amnxn = bm M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42

9 Systems of Linear Equations A number of different things can happen when working with systems of linear equations. A system of linear equations may have: 1 No solution (inconsistent equations), 2 One and only one (unique) solution (consistent equations), or 3 An infinite number of solutions (consistent equations). An important thing to look at when dealing with systems of linear equations is 1 The number of unknowns (variables) and 2 The number of equations. Two Equations in Two Unknowns Three things can happen when working with two equations in two unknowns: no solution, a unique solution, or an infinite number of solutions. Geometrically, what is important to remember is that each equation represents a line in two dimensions. x 2 x 2 x 2 x 1 x 1 x 1 What, if anything, can you say if there are (think visually!): 1 More unknowns than equations, 2 More equations than unknowns. M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 Three Equations in Two Unknowns - Example Three Equations in Three Unknowns Consider the following set of three equations in two unknowns. x1 x2 = 1 x1 + x2 = 2 x1 = 3 x 2 X 1+X 2= X 1=3 x 1 If we encounter this situation, what do we do? Is there a best approximation (solution) to these equations? X 1 X 2=1 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42

10 Two Applications We conclude by briefly introducing two applications that will (hopefully) motivate our looking at solving a linear system. The first example is the problem of balancing chemical equations in chemistry, and The second example is a network flow problem. Balancing Chemical Equations Chemical compounds are represented by chemical formulas, that describe their molecular composition, such as NaCl, HCl, H2O. When chemical compounds are combined under the right combinations, the atoms are rearranged to form new compounds. For example, when methane burns, the methane (CH4) and stable oxygen (O2) react to form carbon dioxide (CO2) and water (H2O). This reaction is represented by the chemical equation CH4 + O2 CO2 + H2O However, this equation does not account for the proportion of molecules required for a complete reaction. The chemical equation is said to be balanced if for each type of atom in the reaction, the same number of atoms appear on each side of the equation, CH4 + 2O2 CO2 + 2H2O M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 Balancing Chemical Equations Example Balance the following chemical equation: HCl + [hydrochloric acid] Na3PO4 [sodium phosphate] H3PO4 + NaCl [phosphoric acid] [sodium chloride] Let x1, x2, x3, and x4 be the positive integers that balance the equation x1(hcl) + x2(na3po4) = x3(h3po4) + x4(nacl) Equating the number of atoms of each type on both sides of the equation we have x1 3x3 = 0 Hydrogen (H) x1 x4 = 0 Chlorine (Cl) 3x2 x4 = 0 Sodium (Na) x2 x3 = 0 Phosphorus (P) 4x2 4x3 = 0 Oxygen (O) Application - Network Flow A network appears in a variety of applications. A network is a set of branches through which something flows. These branches may be wires in electrical circuits through which electricity flows, pipes through which oil or water flows, or street through which cars flow. In many networks, these branches meet at nodes (junctions) where the flow divides. Page 1 of 1 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42

11 Application - Network Flow Example: A Network of Underground Pipes In network analysis, there is generally some measure used to define the rate of flow through the branches, and in many applications there is a conservation of flow so that whatever flows into a node will flow out. A common problem in network analysis is to use known flow rates at certain points in the network to find flow rates in all of the branches. These problems may be solved using linear algebra. Pipes meet at intersections where houses and businesses connect to the system Tyson Street Each section has a flow rate measured in cubic liters per day. Given the flow rates into the network, the rates of flow in each section can be found by solving a set of linear equations. Moultrie Street X Annlin Avenue With flows around the network labeled clockwise from the top left by xi, then beginning at the upper left node, we have x1 + x20 = x1 x2 = x2 x3 = Webster Place. M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 M.Hayes (CAU-GT) Lecture # 1 March 4, / 42 a) Considering the flows given, find the flow in each section of pipe in the network. b) Suppose that a water main breaks at the spot marked with the X. How does this event affect the rates of flow in the other pipes in the network? 4