Linear Algebra Instructor: Justin Ryan ryan@math.wichita.edu Department of Mathematics, Statistics, and Physics Wichita State University Wichita, Kansas Summer 2014 DRAFT 3 June 2014
Preface These lecture notes represent an 8 week course in Linear Algebra. They are meant to supplement the text by Leon. iii
iv
Math 511: Linear Algebra Course Syllabus Summer 2014 Instructor: Justin Ryan CRN: 32251 Office: JB 325 Room: JB 372 Phone: (316) 978 5157 Time: 12:10 1:10 Daily Email: ryan@math.wichita.edu Office: 11:00 12:00 TR Webpage: www.math.wichita.edu/ ryan Textbook: Linear Algebra with Applications, 8 th ed., by Steven J. Leon. Other Materials: Students are not allowed to use calculators, or any other devices, on exams, but may use any resources they like while doing other work for this course. The problems on exams will be structured so that a calculator is not necessary. Attendance: Attendance is required and expected. Students who miss class for any reason are responsible for the material covered that day, and any work that is due. Late work will not be accepted after it has been handed back to the rest of the class. In case you must miss an exam, you will be allowed to make it up before it is returned to the other students in class (assume that this will be the very next class). You must contact me as soon as possible (email is best) to notify me of your intent to take the exam. It will then be available for you to take in the Testing Center in Grace Wilkie Hall. The Testing Center charges a fee of $10. Course Outline: This course will cover vector spaces, linear transformations, inner product spaces, and the theory of eigenvalues in detail. If there is time, we could also briefly study canonical forms or some numerical methods, depending on the interests of the class as a whole. We will use the required text as the main reference for this course, and problems will be assigned from it regularly. However, we will not follow the text in order. We will begin in Chapters 3 and 4, filling in details from Chapters 1 and 2 as necessary. We will then study the second half of Chapter 5, and the first half of Chapter 6. While this may seem rather odd, I believe that this course v
structure will result in a better overall understanding of linear algebra for the students than simply following the book linearly. (Ironic, right?) Recommended Exercises: For each lecture, I will post 10-15 recommended problems on the course webpage (not blackboard!). The only way to learn mathematics is to do it yourself, so it will be very important to complete these. All of these exercises are for your own benefit, and will never be collected. Some of the problems will be similar to examples presented in class; others will be brand new. You should expect exam questions to be similar to these recommended exercises. Good Problems: In addition to recommended exercises, each week you will be assigned a set of Good Problems. These assignments will consists of 5 10 problems that will be collected, and graded based on quality of work and presentation. I will hand out a guideline to completing the good problems before the first set is due. These good problems will be based on the recommended exercises, but may be difficult to complete without doing the REs first. Exams: There will be two (2) midterm exams accounting for 40 % of your final grade, and a comprehensive final exam worth 30 %. You will be allowed to use a 3 5 note card on each exam. The questions will be similar to the recommended exercises. We ll discuss this more before the first exam. The exam schedule for this summer is as follows: Exam Date Tentative Sections Covered Midterm 1 Friday, 13 June 3.1 3.4 Midterm 2 Thursday, 3 July 3.5, 3.6, 4.1 4.3 Final Exam, pt 1 Thursday, 24 July Chs 3 4, 5.3 5.6 Final Exam, pt 2 Friday, 25 July Chs 3 4, 6.1 6.6 Project: Finally, each student will be required to complete a project. Students will have the option to choose between a pure theoretical project or an applied project. Thus each student will be able to tailor at least a portion of the course to their own interests. Further liberties of choice could be given for the project, depending on the class s overall progress in the course. We ll discuss this more around the midpoint of the semester. vi
Grading: Your grade will be determined as follows: Good Problems (8) 1.25 % each Midterm Exams (2) 20 % each Final Project 20 % Final Exam 30 % Your final letter grade will be based on the following scale: 90 100% A 76 78% C+ 88 90% A 68 76% C 86 88% B+ 60 68% D 80 86% B < 60% F 78 80% B There will be no extra credit, and I will not grade on a curve. Academic Honesty: Cheating will not be tolerated. Read the Student Handbook for Wichita State University s official cheating policy. Special Needs: If you have any disability that may impact your ability to carry out any assigned course work in the time allotted, contact the Office of Disability Services (DS), Grace Wilkie Annex, room 173, 978 3309. Assistance: I strongly believe that it is beneficial for students to work together on recommended exercises, good problems, and in preparation for exams. However, it is only beneficial to those students who put in the effort to learn and understand the material; e.g., copying solutions to the good problems from a friend will not help you do well on the exams. Credit Hours: This is a 3 credit hour class. Success in this course is based on the expectation that students will spend a minimum of 135 hours over the length of the course for instruction, preparation, studying, and/or course related activities. This amounts to a minimum of approximately twelve (12) hours per week outside of class. vii
viii
Math 511: Linear Algebra Tentative Course Calendar Summer 2014 This is a tentative outline of what we ll cover in this course, and the order that we will cover it. Exam dates are fixed, but the material that each exam covers could change. The only way to know for sure what an exam will cover is to attend class regularly. Monday Tuesday Wednesday Thursday Friday 2 Intro., 3 3.1 4 3.2 5 3.3 6 3.3 3.1 9 3.4 10 3.4 11 3.4 12 4.1 13 Midterm Exam 1 16 4.1 17 4.2 18 3.5 19 3.5, 4.3 20 4.3 23 3.6 24 3.6 25 Ch 1 4 26 Ch 1 4 27 Ch 1 4 30 5.4 1 5.4 2 5.5 3 Midterm 4 No Exam 2 Class 7 5.5, 5.6 8 5.6 9 5.3 10 5.7 11 6.1 14 6.1 15 6.2 16 6.3 17 6.5? 18 6.6? 21 Ch 9? 22 Ch 9? 23 Final 24 Final 25 Final Project Exam pt 1 Exam pt 2 Notice that no sections from chapters 1 or 2 are explicitly listed in the calendar. This does not mean that we won t cover them. In fact, we will cover everything from chapters 1 and 2 as examples, exercises, and methods of solving problems in chapters 3 and 4. Thus, we should be finished with all of chapters 1, 2, 3, and 4 in the first 4 weeks of the semester. Again, this schedule is subject to change based on how the semester goes. However, everything in red should be considered fixed. ix
x
Contents Preface Course Syllabus Course Calendar iii v ix 1 Vector Spaces 1 Lecture 1..................................... 1 Lecture 2..................................... 5 Appendix 11 Good Problems................................. 13 Midterm Exams................................. 14 Final Projects.................................. 15 Final Exams................................... 16 xi
xii
1 Vector Spaces We begin our study of linear algebra in chapter 3 of the text, but we ll also spend a lot of time reviewing the ideas of Chapters 1 and 2. Lecture 1 This lecture covers section 3.1 of the text. Let V be a set on which operations of addition and scalar multiplication are defined. We mean this in the following abstract sense. Addition takes two elements of V as arguments, and produces another element of V as its output. This situation is denoted symbolically by + : V V V : (x, y) x + y. Similarly, scalar multiplication takes a number (usually a real number) and an element of V as input, and outputs an element of V. It is denoted by : R V V : (α, x) α x = αx. It is very important that the images of these operations lands back in V. A set V is said to be closed with respect to an operation if and only if for each input, there is a unique output in the set V. This does not necessarily happen for every operation. Consider the following example. Example 1 Let V = Z be the integers. Define addition and scalar multiplication by real numbers in the usual ways. Observe that Z is closed with respect to addition, but not closed with respect to scalar multiplication by real numbers. 1
We can now define a vector space, the main object of study in this course. Definition 2 Let V be a set on which the operations of addition (+) and scalar {vsd} multiplication ( ) are defined. Suppose further that V is closed with respect to both operations; that is, suppose (V,+, ) satisfies C1. If x V and α R, then αx V ; and C2. If x, y V, then x + y V. The set (V, +, ) with operations of addition and scalar multiplication satisfying C1 and C2 is said to form a (real) vector space if and only if the following axioms are satisfied: A1. x + y = y + x for any x, y V ; A2. (x + y) + z = x + (y + z) for any x, y, z V ; A3. There exists an element 0 V such that x + 0 = x for each x V ; A4. For each x V, there exists an element x V such that x + ( x) = 0; A5. α(x + y) = αx + αy for every α R, and every x, y V ; A6. (α + β)x = αx + βx for every α,β R and any x V ; A7. (αβ)x = α(βx) for every α,β R and any x V ; and A8. 1 x = x for all x V. The set V is called the universal set of the vector space. Elements of V are called vectors in the vector space. Remark 3 For most (possibly all) of this class, we will only consider real vector spaces. However, definition 2 can be modified to define a complex vector space by simply swapping out all of the R s for C s. This definition works perfectly fine as an abstract description of a vector space. However, these ten axioms can be quite intimidating, especially if you re not used to thinking in such an abstract setting. In order to build our intuition, and hopefully become more comfortable with this definition, we will now spend a significant amount of time looking at examples of vector spaces. 2
Example 4 The vector spaces R and R 2. Example 5 The vector spaces R n. 3
4
Lecture 2 We shall continue to look at examples of vector spaces, then prove some properties of vector spaces that arise as consequences of the abstract definition. Example 6 The real vector space C. Example 7 The complex vector space C. 5
Example 8 The vector space of m n matrices, R m n. 6
Example 9 (3.1.6!) The vector space P n. Example 10 (3.1.16!!) An isomorphism between P n and R n. 7
Example 11 (3.1.5!) The vector space C [a, b] of continuous functions on a closed interval [a,b]. Example 12 The vector spaces C r [a,b]. 8
Now that we have an arsenal of examples to keep in mind, let s look at some theoretical consequences of Definition 2. Theorem 13 If V is a vector space and x V, then 1. 0x = 0; 2. x + y = 0 implies that y = x; i.e., the additive inverse of x is unique (there is only one); 3. ( 1)x = x. Proof: 9
Example 14 (3.1.7) Show that the 0 element of a vector space is unique. Proof: Example 15 (3.1.8) Let V be a vector space and x, y, z V. Prove that if x + y = x + z, then y = z. This is known as the Cancellation Law. Proof: 10
Appendix Blank copies of Good Problems, Midterm Exams, Final Project, and the Final Exam will be collected in this appendix. Other assignments and handouts may be included here as well. 11
12
Good Problem sets will go here. 13
Midterm Exams will go here. 14
Final Projects will go here. 15
Final Exams will go here. 16