# MATH 341: Linear Algebra and Matrix Theory

### General Information

```      Instructor: Dave Rusin (rusin@math.utexas.edu)
Office hrs: I will be in my office (RLM 9.140) at these times for you:
Tuesdays 11-12:15, and 2-5 pm
Wednesdays noon-3pm
Thursdays 11-12:15 and 2-3 pm
* Warning: hours may well get tweaked early in the semester *

Text: Andrilli & Hecker, Elementary Linear Algebra (fifth edition)

This class (UniqID 54190) meets T,Th 12:30-3:00pm in RLM 5.122

Teaching asst: Spencer Johnson

Your final exam is Wednesday, December 19, 2:00-5:00 pm. It may not be
in the regular classroom; I will announce the location when I know it.
There is no provision for taking the final exam earlier or later.
```

Course webpage: http://www.ma.utexas.edu/~rusin/341-18b/ It is unlikely that I will post any important material to Canvas; for any additional information I want to give you outside of class you should come to this webpage.

NOVEMBER UPDATE

There will be no class on Tuesday, November 20.

Here is a homework assignment due THURSDAY, Nov 29:

``` section 5.1 # 1e, 1g, 5, 7, 16, 19
section 5.2 # 2d, 3d, 7b, 14
```

### Catalogue Description

Vector spaces, linear transformations, matrices, linear equations, determinants. Some emphasis on rigor and proofs.

The emphasis in this course is on understanding the concepts and learning to use the tools of linear algebra and matrices. The course is proof-based. The fundamental concepts and tools of the subject covered are:

• Matrices: matrix operations, the rules of matrix algebra, invertible matrices.
• Linear equations: row operations and row equivalence; elementary matrices; solving ystems of linear equations by Gaussian elimination; inverting a matrix with the aid of row operations.
• Vector spaces:vector spaces and subspaces; linear independence and span of a set of vectors, basis and dimension; the standard bases for common vector spaces.
• Inner product spaces: Cauchy-Schwarz inequality, orthonormal bases, the Gramm-Schmidt procedure, orthogonal complement of a subspace, orthogonal projection.
• Linear Transformations: kernel and range of a linear transformation, the Rank- Nullity Theorem, linear transformations and matrices, change of basis, similarity of matrices.
• Determinants: the definition and basic properties of determinants, Cramers rule.
• Eigenvalues: eigenvalues and eigenvectors, diagonalizability of a real symmetric matrix, canonical forms.

### Pre-requisites

Mathematics 408D with a grade of at least C-

There are two Linear Algebra courses at UT, Math 340L and Math 341, which are fairly similar. You cannot earn UT credit for both of them. Math 341 is ordinarily limited to math majors, and Math 340L is more appropriate for non-math majors. Please see an advisor in MPAA (on the ground floor of RLM) if you need assistance enrolling in the appropriate Linear Algebra course.

A strong mathematics major should take Mathematics 341 immediately after M408D; those who are unready to take a proof-based course should take M325K or a similar course first.

Homework: I will assign homework problems, typically taken from the book, approximately weekly. There will be a grader to try to get as much of your responses graded as possible but I strongly encourage you to self-grade, that is, consult with me or your classmates to know that your answers are good. Remember, you do homework primarily to learn the material, not to score points.

I will give a grade for each homework set, then drop the lowest two, then scale your remaining total to a 100-point scale as part of your semester grade.

Please note that the answer to a homework question will never be something like a "6" with a big circle around it. Most of your questions will come to you in the form of words, and you will give me back a few sentences of response. So your answers are not necessarily called "correct" or "incorrect"; there are "good" responses and "less good" ones. We will spend the semester trying to get used to the good ones!

Exams: There will be 2 mid-term exams, to be held during the usual class period, and a comprehensive final exam. I expect the midterm dates to be October 2 and November 8 but the actual dates will be announced in mid-September.

Please mark on your calendars now the time and date of the final exam. (I don't know yet what room the final exam will be held in.) Textbooks, notes, and electronic devices (including phones and calculators) are not permitted during exams. The exams will be a mix of multiple-choice and free-response questions; the ratio will change as the semester progresses.

You semester grade will be a weighted average of these components: your homework average, two mid-term exams, and the final exam which counts double. These will be letter grades, which I average the way GPAs are averaged. For example, if a student's grades on these four components are B+, C+, A-, and B respectively, their semester grade on a 4.00 scale will be

```   0.2 x (3.33) + 0.2 x (2.33) + 0.2 x (3.67) + 0.4 x (3.00) = 3.066
```
which is closer to a B (3.00) than to a B+ (3.33) so the student's semester grade is a B.

The reason I average letter grades is to give you the benefit of the doubt on each component. Nominally your letter grade on any component is the familiar high school scale:
 97-100 A+ 94-96 A 90-93 A- 87-89 B+ 84-86 B 80-83 B- 77-79 C+ 74-76 C 70-73 C- 67-69 D+ 64-66 D 60-63 D- 0-59 F
But I recognize that for example I may inadvertently give an exam which is exceptionally hard, and I don't want to penalize the whole class. So I will compute your letter grade on that component a different way too, and then the grade I actually record will be the higher of the two (the straight scale above or this curved scale). Here is how the curved scale works. I will compute the mean and standard deviation of the class's raw scores. The mean will become the center of the "B" range, and from there, each possible letter grade is a range whose width is exactly 1/3 of the standard deviation.

For example, suppose the class average on the exam was 85.5 points and the standard deviation was 14.4 points. Then the conversion from raw scores to letter grades will be based on these brackets (of width 14.4/3 = 4.8 centered at values 85.5 + 4.8n) :
 SAMPLE! 103+ A+ 98-102 A 93-77 A- 88-92 B+ 84-87 B 79-83 B- 74-78 C+ 69-73 C 64-68 C- 60-63 D+ 55-59 D 50-54 D- 0-49 F
Note that in this example any student whose raw score was 87 or higher would get a higher letter grade based on the traditional 90-80-70-60 scale shown earlier, and thus for those students the traditional scale will be used. Everyone else would benefit from the curved scale, and so for these students I the second scale would be the one that gives them their grade.

I'll do this computation separately for each exam and for the homework total.

### Policies

Classroom activity: Our meeting times together are very short so we must make the most of them. Come to class daily and ask questions; this is greatly facilitated by reading ahead each day and doing the homework problems as they are assigned. Please silence your cell phones. I will always assume that any conversations I hear are about the course material so I may ask you to speak up.

Textbooks, notes, and electronic devices (including phones and calculators) are not permitted during exams.

Make-ups: it is in general not possible to make up missing quizzes or homework assignments after the due date. If you believe you will have to miss a graded event, please notify me in advance; I will try to arrange for you to complete the work early.

Students with disabilities: The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities. For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.

Religious holidays: If you are unable to participate in a required class activity (such as an exam) because it conflicts with your religious traditions, please notify me IN ADVANCE and I will make accommodations for you. Typically I will ask you to complete the required work before the religious observance begins.

Academic Integrity. Please read the message about Academic Integrity from the Dean of Students Office. I very much prefer to treat you as professionals whose honesty is beyond question; but if my trust is violated I will follow the procedures available to me to see that dishonesty is exposed and punished.

Campus safety: Please familiarize yourself with the Emergency Preparedness instructions provided by the university's Campus Safety and Security office. In the event of severe weather or a security threat, we will immediately suspend class and follow the instructions given. You may wish to sign up with the campus alert programs.

Counseling: Students often encounter non-academic difficulties during the semester, including stresses from family, health issues, and lifestyle choices. I am not trained to help you with these but do encourage you to take advantage of the Counselling and Mental Health Center, Student Services Bldg (SSB), 5th Floor, open M-F 8am-5pm. (512 471 3515, or www.cmhc.utexas.edu

Add dates: If you enroll within the first four class days of the semester, and have missed any graded material, I will adjust the weighting of your graded sections accordingly so that you are not penalized. No such accommodation is made for students who enroll on the 5th day or later. (Such students must enroll through the MPAA advising center in RLM, and ordinarily I do not admit students who ask to enroll then if they have missed any graded activities).

Drop dates: Sept 4 is the last day to drop without approval of the department chair; Sept 14 is the last day to drop the course for a possible refund; Nov 1 is the last day an undergraduate student may, with the dean's approval, withdraw from the University or drop a class except for urgent and substantiated, nonacademic reasons. For more information about deadlines for adding and dropping the course under different circumstances, please consult the Registrar's web page, http://registrar.utexas.edu/calendars/18-19/

Computers: We don't make use of sophisticated software in this class, but if you find this interesting, you are welcome to use the department's computer facilities. Our 40-seat undergrad computer lab in RLM 7.122, is open to all students enrolled in Math courses. Students can sign up for an individual account themselves in the computer lab using their UT EID. We have most of the mainstream commercial math software: Mathematica, Maple, Matlab, etc., and an asortment of open source programs. If you come to my office you will see me use some of this software to help illustrate concepts. Please see me if you would like more information.

Bennett exam: Starting last year, the U.T. Mathematics Department started an annual competition in Linear Algebra, the Bennett contest exam. This is open only to students who have completed a U.T. Linear Algebra course during 2018 -- and that means you! The questions are based on the topics covered in this course, but require more than the usual amount of persistence and cleverness. There are cash prizes for the top scorers. Please plan to participate! This year's contest is scheduled for Tuesday, December 11.

### Schedule

The following table is a tentative schedule for the course. Please be aware that material may be reordered, added or deleted.

• 1 Vectors and matrices (2 weeks)

• 1.1 Vector addition, scaling, dot product

• 1.2 Cauchy-Schwarz and Triangle inequalities

• 1.4,1.5 Matrix addition, scaling, multiplication

• 1.3 Proof techniques

• 2 Systems of linear equations (2 weeks)

• 2.1 Solution sets, Gauss elimination

• 2.2 Consistent and inconsistent systems

• 2.2 Row echelon forms, Gauss-Jordan reduction

• 2.3-2.4 Rank, row space and inverse of a matrix

• 3 Determinants and eigenvalues (2 weeks)

• 3.1 Determinant of a matrix, properties

• 3.1 Cofactors and adjoint of a matrix

• 3.3 Matrix inverse formula, Cramer's rule

• 3.4 Eigenvalues and eigenvectors

• 3.4 Diagonal factorization of a matrix

• 4 Vector spaces (3 weeks)

• 4.1-4.2 Idea of vector spaces, subspaces

• 4.3-4.4 Span and linear independence of vectors

• 4.5 Bases and dimension of vector spaces

• 4.6-4.7 Coordinates with respect to a basis

• 4.7 Change of basis formula for vectors

• 5 Linear transformations (2 weeks)

• 5.1 Idea of a linear transformation

• 5.2 Matrix representation

• 5.3 Kernel and range, Dimension Theorem

• 5.4-5.5 One-to-one, onto, and isomorphisms

• 5.6 Eigenvalues, eigenvectors and diagonalization

• 6 Orthogonality (2 weeks)

• 6.1 Orthonormal bases, Gram-Schmidt process

• 6.2 Orthogonal complements, Projection Theorem

• 6.2 Orthogonal projections of vectors

• 6.3 Diagonalization of symmetric operators