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MATH 371 Section 001: Advanced Linear Algebra

Fall 2011

3 Credit Hours
Primary Instructor: Dr. Gregory Knese
Syllabus subject to change.
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Class Time and Location

The class is TTh 9:30-10:45am in Gordon Palmer 153.  The final exam is Dec 14, 8-10:30am.

Prerequisites

From the Student Records System
  • MATH 237 (undergrad) with a minimum grade of C-
  • Or
  • MATH 257 (undergrad) with a minimum grade of C-

You are expected to be comfortable with all of the standard matrix algebra tasks: row operations, Gaussian elimination, finding inverses, finding a nullspace and column space, determinants, and basics of eigenvalues/eigenvectors.  We will review most of this from a more abstract viewpoint, and seeing it this way for the first time is usually not ideal. 

Course Description

This is a rigorous course in linear algebra.  We will review many of the topics from matrix algebra using a more abstract approach through vector spaces.  It will be extremely helpful for you to already be comfortable with the computational aspects.  Later on, we study linear transformations in greater depth than you have probably seen.  Two big goals are the spectral theorem and the Jordan canonical form.  

Aside from the actual material of the course, learning to read mathematics (digesting definitions and theorems and understanding proofs) and learning to write mathematics (writing proofs) are the most important parts of the course. 

From time to time we may use the computer program "Sage" for some computations.  (Anyone can download Sage or use it online: http://www.sagemath.org/)

Student Learning Outcomes

  • Understand vector spaces and the important notions of linear combination, basis, dimension 
  • Understand the notion of linear transformation and its correspondence with matrices.
  • Understand inner product spaces and their related operators (normal, self-adjoint, unitary, and orthogonal operators) as well as the spectral theorem
  • Obtain a deeper understanding of eigenvalues/eigenvectors as well as understand the Jordan Canonical form
  • Become proficient in reading and writing rigorous mathematics.

Outline of Topics

Course catalog description: Topics include inner product spaces, norms, self adjoint and normal operators, orthogonal and unitary operators, orthogonal projections and the spectral theorem, bilinear and quadratic forms, generalized eigenvectors, and Jordan canonical form.

Since the prerequisite for this course is either Math 237 or 257 (and I haven't seen 257 offered recently), our plan will be to start from the beginning but assume you know how to do the computations.  

The topics will be:

Chapter 1: vector spaces, subspaces, linear combinations, systems of equations, linear dependence/independence, bases, dimension

Chapter 2: linear transformations, null space, range, matrix representation of a linear transformation, composition of linear transformations, matrix multiplication, invertibility, isomorphisms, change of coordinate matrix

Chapter 3: elementary matrix operations and elementary matrices, rank of a matrix, matrix inverse, systems of linear equations.  (This chapter will probably go quickly.)

Chapter 4: determinants, properties of determinants.  (We will only look at section 4.4 as a refresher.)

Chapter 5: Eigenvalues, eigenvectors, diagonalizability, invariant subspaces, cayley-hamilton theorem

Chapter 6: inner product spaces, gram-schmidt, orthogonal complements, adjoint of a linear operator, normal and self-adjoint operators, unitary and orthogonal operators, orthogonal projections, the spectral theorem.

Chapter 7: the Jordan canonical form, minimal polynomial 

Exams and Assignments

Exams

There will be two exams and a final. All exams are cumulative. Exam #1 is Sept 27 and Exam #2 is Oct 25; both are in class at the regular time. The final exam is Dec 14 (8am-10:30am).

 

Quizzes

Approximately every third class there will be short quiz on definition and theorem statements.  (Theorem statements will be confined to big "named"  theorems; e.g. "the spectral theorem.")

Homework Assignments

There will be weekly written homework assignments.  These should be written up nicely, just as you might write up an English essay.  (It is not necessary to type, but I would encourage you to do so with the typesetting program LaTeX )

You may consult with other students or myself on how to do a problem only after you have given a genuine effort to solve the problem.  After you have solved your homework problems, your solutions must be written completely independently.

 

Grading Policy

Grading breakdown:

Final: 30%

Exams: 15+15=30%

Quizzes: 10%

Homework: 30%

Letter grades will be determined using a the scale: A+ = [97,100], A = [93.5, 97), A- = [90, 93.5), and so on for B,C,D, and F = [0,60)

Policy on Missed Exams & Coursework

Your lowest exam score will be replaced by your final exam score if it is to your benefit.  For this reason, there will be no make-ups for missing an exam: if you miss an exam it will be replaced by your final exam.  No one should miss both exams nor should anyone miss the final.

10% of your quiz grades will be dropped, and therefore no make-ups will be permitted.

10% of your homework scores will be dropped.  

Attendance Policy

You are expected to be prepared for and participate in every class.  Being prepared essentially means doing any assigned reading and recommended problems ahead of time.  This is essential for doing well in this class. Coming to class "cold" is not recommended.

 I will take attendance but it will not be a formal part of your grade.

Required Texts

UA Supply Store Textbook Information

  • FRIEDBERG / LINEAR ALGEBRA
    (Required)
  • FRIEDBERG (RENTAL) / (RENTAL) LINEAR ALGEBRA
    (RENTAL)

Other Course Materials

You may want to take a look at the online book: "Linear Algebra Done Wrong" It covers everything we are covering.

It might be useful to get a book on writing proofs.  There are many books like this: "Transition to Higher Mathematics" by Dumas and McCarthy, "A Transition to Advanced Mathematics" by Smith, Eggen, St. Andre, or "The Nuts and Bolts of Proofs" by Cupillari.  The website http://zimmer.csufresno.edu/~larryc/proofs/proofs.html might also be useful.

The program Sage may be useful for some computations: http://www.sagemath.org/

The website Wolfram Alpha is useful for quick computations or information, but you may need to know what you are doing to interpret what it spits out.

Policy on Academic Misconduct

All students in attendance at the University of Alabama are expected to be honorable and to observe standards of conduct appropriate to a community of scholars. The University expects from its students a higher standard of conduct than the minimum required to avoid discipline. Academic misconduct includes all acts of dishonesty in any academically related matter and any knowing or intentional help or attempt to help, or conspiracy to help, another student.

The Academic Misconduct Disciplinary Policy will be followed in the event of academic misconduct.

Disability Statement

If you are registered with the Office of Disability Services, please make an appointment with me as soon as possible to discuss any course accommodations that may be necessary. If you have a disability, but have not contacted the Office of Disability Services, please call 348-4285 or visit 133-B Martha Parham Hall East to register for services. Students who may need course adaptations because of a disability are welcome to make an appointment to see me during office hours. Students with disabilities must be registered with the Office of Disability Services, 133-B Martha Parham Hall East, before receiving academic adjustments.

Severe Weather Protocol

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