Linear Algebra -- Math 544

Frank Thorne - Fall 2015

University of South Carolina

Instructor: Frank Thorne, LeConte 317O, thorne [at] math [dot] sc [dot] edu

Office Hours: Tuesdays and Wednesdays, 4:00-5:00 and Thursdays, 11:15-12:15.

Course objectives/learning outcomes:

Successful students will:
• Understand the concept of a real vector space from an algebraic and a geometric point of view. The student will master related concepts such as bases and linear independence, norms, inner products, and so forth.
• Understand the most important maps between vector spaces -- the linear transformations. The successful student will be able to describe their structure and describe it in terms of ranks, nullspaces, etc.
• Understand how to manipulate matrices and explain the correspondence between matrices and linear transformations.
• Understand what eigenvalues are, why they are interesting, and why one would want to compute them.
• See applications of all of the above to engineering and physical and social sciences. To give just one example --- at the heart of Google search is an eigenvalue. (The student will not master this over the course of only one semester! If you give it thirty years, maybe. But people are developing new applications for linear algebra all the time.)
• Develop their ability to understand and explain rigorous mathematics and prove simple statements. This is a 500-level math class, and as such proofs will be required on homeworks and exams. But I understand that many of you have very limited background in this.

Course non-objectives/learning non-outcomes:

Linear algebra is too big of a subject to really do justice to in one semester. Therefore we will mostly omit some beautiful topics.

• The course will say little about the very interesting and important questions when doing linear algebra in the "real world" -- which involves imprecise measurements and computational precision issues. The student who would like to see this beautiful topic emphasized should instead take Math 526.
• The course will develop only a minimal amount of formalism -- so the vector space axioms, fields other than R, infinite-dimensional vector spaces, inner products other than the "obvious" one coming from geometry, etc. will not be talked about much. This material would be more appropriate if the course had prerequisites such as Math 300 and Math 574. Please see me at the end of the term if you would be interested in learning more about this aspect of the subject.

Warning. You should expect 5-8 hours of homework a week in this class, which is more than most other instructors assign; in my experience there is no other way to learn the material. Your consistent effort will certainly lead to improved understanding, and it will almost certainly lead to you earning high grades.

• Text : Knop, Linear Algebra: A First Course With Applications, buy it directly from the publisher or at the bookstore or elsewhere. The publisher is offering a 20% discount and free shipping through the end of the term if you order directly from them.

Note that you should plan on thoroughly reading the book. I spent a lot of effort picking one that doesn't suck. Just going to the lectures without reading will likely be insufficient to master the course material.

• Lectures : LeConte 115, MWF, 9:40-10:30 am.

• Exam schedule : In-class midterms will be on Wednesday, October 7 Wednesday, October 14 and Wednesday, November 11 Monday, November 23.

The final exam will on Friday, December 11 at 9:00 a.m. (I will bring coffee to share.)

• Homework and Quizzes :

Warning. I assign a lot of homework.

The homework is intended to take 5-8 hours a week. That is a lot. Please count on making a consistent effort to do well in this class! Starting the night before is a bad idea.

If homework takes you more than 10 hours on any given week, then that is more than I intended; please let me know.

There will be at least one bonus problem on each homework. This is the one and only way you can earn extra credit.

Quizzes will be given each Monday. All quiz problems will be extremely similar to (or taken verbatim from) assigned homework exercises.

Please note. You will be graded both on correctness and on quality of exposition. Indeed, a major focus of Math 544 is the ability to communicate mathematical ideas clearly. The standard is that someone who doesn't know the answer should be able to easily follow your work. In particular, please write in complete English sentences and draw clear diagrams where appropriate. Any work that is confusing, ambiguous, or poorly explained will not receive full credit.

Grading scale: For the homework/quiz, A = 90+, B+ = 85+, B = 78+, C+ = 73+, C = 66+, D = 52+.

For the first midterm, A = 86+, B = 68+, C = 52+, D = 42+.

For future exams, the grading scale will be at least as generous as that for the homework/quizzes, and perhaps more so if the difficulty of the exam warrants it.

Please note that my grading scale is more generous than the usual 10-point scale. However, I am a (slightly) stricter grader than most. This is intended to balance out.

 Grade component % of grade Two in-class exams 20% x 2 Final exam: 35% Homework and Quizzes: 25%

Please note: If you come to office hours, please don't be shy about interrupting me! I'm usually in the middle of something, but it can wait.

• Make-up policy :

If you have a legitimate conflict with any of the exams it is your responsibility to inform me at least a week before the exam. Otherwise makeup exams will be given only in case of documented emergency.

Late homework will not be accepted, and makeup quizzes will not be given, except in cases of medical emergency. Instead, your lowest two grades will be dropped.

Academic honesty and attendance are expected of all students.

Calculators will not be needed or allowed.

• Schedule and homeworks :

• Due August 31:

1.1: A 5, 11, 16; B 1, 4, 8.

1.2: 1(c-d), 6(a-b), 7(a-b), 10(a-b), 11(a-b), 12(a-b); B 1, 4, 9, 11.

1.3: A 1(a-b), 18; B 2, 7.

1.4: A 5(a-b), 7(a-b), 11; B 11, 12.

• Quiz 1, with solutions to quiz and 1.1 B4, 1.4 A7(a-b).

• Due Wednesday, September 9:

1.5: A 1, 5, 9, 14, 20. (Please draw schematic pictures to accompany your solutions!)

1.5: B 1, 3, 4, 5, 9.

2.1: A 1(a-b), 5(b), 7(d).

Quiz 2, with solutions.

• Due Monday, September 21 (not September 14):

2.3: A 1, 3, 4; B 1-8, 11.

2.4: A 1, 6, 8, 17; B 1, 4, 6, 7.

2.5: A 1, 5, 11; B 1, 2, 7.

Quiz 3, with solutions.

• Due Monday, September 28:

2.5: A 9, 12; B 10, 11.

3.1: A 5 (d-f), 8, 9; B 2(d-f), 3(a-b), 4(a-b), 7.

3.2: 5(a-c), 10, 11, 14; B 3, 6, 8, 9.

Bonus: 3.2, B, 16.

Instructions: When describing elementary row operations, just say what you are doing; never mind about the "type".

Quiz 4, with solutions.

• Due Monday, October 5 Monday, October 12: (Campus closed October 5-9 due to flooding.)

4.1: A 3, 5, 7, 8; B 1, 2, 6, 10, 15.

4.2: A 1, 2, 7, 8; B 1, 4.

Bonus: Find a polynomial p(t) such that p(n) and the derivative p'(n) are both equal to n for n = 1, 2, 3, 4, and 5. You are welcome to use a calculator or computer. If so, your solution should say what you asked the computer to compute and what the answer was.

• Exam 1 was on Wednesday, October 14. Exam and solutions.

Quiz 5, with solutions.

Homework 6, due Monday, October 26.

Lecture notes from October 19.

Quiz 6, with solutions.

Homework 7, due Monday, November 2.

Quiz 7, with solutions.

Lecture notes through October 28.

Study guide for Midterm 2.

• Due Monday, November 9:

4.4: A 1, 4, 7; B 1, 3, 5, 11; 4.5: A 1, 2, 6, 8, 9; B 2, 7, 10.

Quiz 8, with solutions.

• Due Friday, November 20:

4.6: A 2, 6, 8, B 1, 4, 5, 7; 6.2: A 1, 6. 7; B 2, 3, 5.

Quiz 10, with solutions.

• Exam 2 was on Monday, November 23. Exam and solutions.