MATH 245: Linear Algebra 2 (Advanced Level)

SPRING 2018

• Lectures: Mon/Wed/Fri 9:30am–10:20am in QNC 1507.

• Instructor: Spiro Karigiannis Email: karigiannis@uwaterloo.ca

• Telephone: 519-888-4567 ext 32810 Office: MC 5326

• Office Hours: Mon 1:00pm–2:00pm; Wed 1:00pm–2:00pm; or by appointment.

• Midterm Test: Tuesday, June 5, 4:30pm-6:20pm, in MC 2034.

• Course Website: https://learn.uwaterloo.ca/

• The course website is on UW-LEARN and all materials (assignments, solutions, supplementary material) will beposted there. You will need your UW login ID to access the course website. It is your responsibility to check thecourse website on a regular basis. Ignorance is not an acceptable excuse for missing deadlines.

• All assignments must be submitted via Crowdmark, and all graded work (assignments and midterm) will be returnedto students via Crowdmark.

Course Description: The calendar description of this course is “An advanced level version of MATH 235”. Thedescription of MATH 235 is:

Orthogonal and unitary matrices and transformations. Orthogonal projections, Gram-Schmidt procedure, best approxima-tions, least-squares. Inner products, angles and orthogonality, orthogonal diagonalization, singular value decomposition,applications.

MATH 245 is in fact much, much more than just an advanced level version of MATH 235. We will cover all of theabove topics, in much more generality and abstraction, in about half of the course. We will actually begin with muchdeeper results on the structure of linear operators on a finite-dimensional vector space over any field, namely the primarydecomposition theorem, the cyclic decomposition theorem, and their consequences such as the rational and Jordancanonical forms. This first half of the course is not covered at all in MATH 235. Then we will study bilinear forms overany field, including the canonical form theorems for symmetric and skew-symmetric bilinear forms. This topic also is notcovered at all in MATH 235. Finally, we will get to inner product spaces; unitary, self-adjoint, and normal operators;and the spectral theorem. We will cover these topics in much greater depth than MATH 235.

We will then discuss the tensor product V ⊗W of vector spaces, and the exterior algebra Λ•(V ) of a vector space, and usethe exterior algebra to give a basis-free invariant construction of the determinant and the other elementary symmetricpolynomials associated to a linear operator. Another possible additional topic is complex structures on real vector spacesand their relation with complexification.

Brief description of topics:

• the algebra of polynomials; Lagrange interpolation; polynomial ideals; prime factorization of a polynomial

• the minimal polynomial; T -invariant subspaces; T -conductors

• characterization of diagonalizability and triangularizability in terms of minimal polynomial

• the primary decomposition theorem

• cyclic subspaces and annihilators; the cyclic decomposition theorem and the rational canonical form

• the Jordan form; computation of invariant factors; semi-simple operators

• symmetric and skew-symmetric bilinear forms; canonical form theorems; groups preserving bilinear forms

• inner product spaces; adjoints; unitary operators; self-adjoint opertors; normal operators

• forms on inner product spaces; positive forms; spectral theory

• canonical isomorphisms induced by nondegenerate bilinear forms

• possible additional topics: quotient spaces; exterior algebra; tensor products; elementary symmetric polynomials;complexification; complex structures on real vector spaces

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Prerequisites: The course prerequisite is Math 146 [Linear Algebra 1 (Advanced Level)]. This course is for HonoursMathematics students only. You are expected to have had significant previous exposure to rigourous mathematical proofs.We will cover a large amount of difficult material at a very fast pace.

Textbook: There really is only one textbook which is ideally suited for this course:

“Linear Algebra: Second Edition” by Kenneth Hoffman and Ray Kunze; Prentice Hall, 1971.

It is in my opinion the best written, most complete textbook on advanced linear algebra in existence. In particular, itvery carefully treats the primary decomposition theorem, the cyclic decomposition theorem, and the rational and Jordancanonical forms. It also includes a nice discussion of the Grassman algebra of a vector space, which is a topic that Iintend to cover. However, this book is out of print. The UW Bookstore has obtained permission from the publisher toprint copies (in the form of course notes) that will be available for purchase. I will be following this book quite closely,for most of the course. A copy will also be put on reserve in the Davis library. Some other useful textbooks for thiscourse are:

• “Linear Algebra Done Right” by Sheldon Axler; Springer, 2015.

• “Finite-Dimensional Vector Spaces” by Paul Halmos; Springer, 1987.

• “Linear Algebra: Fourth Edition” by Friedberg, Insel, and Spence; Prentice-Hall, 2003.

These four books will all be on reserve in the Davis library. If you attend all of the lectures (as you should) then you canprobably get by without needing a textbook.

Marking scheme: Your course mark will be determined as follows:

• Assignments: 30% (twelve assignments, one every week, worth 2.5% each)• Midterm test: 15% (on Tuesday, June 5, 4:30pm-6:20pm, in MC 2034)• FINAL EXAM: 55% (date, time, and location TBA)

You may work together with your classmates on the assignment problems, you must write up and turn in your ownsolutions to the problems. Be sure to state which classmates you worked with on your assignment. The assignments arean integral part of your evaluation in this course and I encourage everyone to take them very seriously. I will not besympathetic to requests for leniency after the exam if you have not done the assignments.There will not be an opportunity to base your entire course mark on the final exam. This course willcover a large amount of difficult material at a fast pace. If you do not work hard in this course from dayone, you will not do well.

There will be no opportunity for a make-up midterm test. A student who misses the midterm test without a valid,acceptable excuse (accompanied by documented proof, such as a medical note) will receive a score of zero on the test.Students who miss the midterm for valid reasons will have the 15 points missed transferred to the final exam.

Assignment due dates:

• Assignment 01.....due Wednesday, 09 May, 2018.• Assignment 02.....due Wednesday, 16 May, 2018.• Assignment 03.....due Wednesday, 23 May, 2018.• Assignment 04.....due Wednesday, 30 May, 2018.• Assignment 05.....due Wednesday, 06 June, 2018.• Assignment 06.....due Wednesday, 13 June 2018.

• Assignment 07.....due Wednesday, 20 June, 2018.• Assignment 08.....due Wednesday, 27 June, 2018.• Assignment 09.....due Wednesday, 04 July, 2018.• Assignment 10....due Wednesday, 11 July, 2018.• Assignment 11....due Wednesday, 18 July, 2018.• Assignment 12....due Wednesday, 25 July, 2018.

Assignments are due by 11:59pm on the due date. Late assignments will be automatically given a grade of zero byCrowdmark. DO NOT WAIT UNTIL THE LAST FEW MINUTES TO SUBMIT YOUR ASSIGNMENT!Problems or delays with internet service and/or the Crowdmark website are not acceptable excuses for late submission.

IMPORTANT: Any concerns with marking on assignments or midterms should be brought to my attention no morethan one week (seven days) after that assessment has been returned to you. Any re-marking requests made after thisseven day period will not be considered. Moreover, re-marking requests cannot be made via email. They must be madeonly in person, such as during office hours.

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Academic offenses

Academic Integrity: In order to maintain a culture of academic integrity, members of the University of Waterloo commu-nity are expected to promote honesty, trust, fairness, respect and responsibility. Please see http://www.uwaterloo.ca/

academicintegrity/ for more information.

Grievance: A student who believes that a decision affecting some aspect of his/her university life has been unfair orunreasonable may have grounds for initiating a grievance. Read Policy 70 - Student Petitions and Grievances, Section4, http://www.adm.uwaterloo.ca/infosec/Policies/policy70.htm. When in doubt please be certain to contact thedepartments administrative assistant who will provide further assistance.

Discipline: A student is expected to know what constitutes academic integrity, to avoid committing academic offenses,and to take responsibility for his/her actions. A student who is unsure whether an action constitutes an offense, orwho needs help in learning how to avoid offenses (e.g., plagiarism, cheating) or about rules for group work/collaborationshould seek guidance from the course professor, academic advisor, or the Undergraduate Associate Dean. For informationon categories of o?enses and types of penalties, students should refer to Policy 71, Student Discipline, http://www.adm.uwaterloo.ca/infosec/Policies/policy71.htm. For typical penalties check Guidelines for the Assessment of Penalties,http://www.adm.uwaterloo.ca/infosec/guidelines/penaltyguidelines.htm.

Avoiding Academic Offenses: Most students are unaware of the line between acceptable and unacceptable academic be-haviour, especially when discussing assignments with classmates and using the work of other students. For information oncommonly misunderstood academic offenses and how to avoid them, students should refer to the Faculty of MathematicsCheating and Student Academic Discipline Policy, http://www.math.uwaterloo.ca/navigation/Current/cheating_policy.shtml

Appeals: A student may appeal the finding and/or penalty in a decision made under Policy 70 - Student Petitions andGrievances (other than regarding a petition) or Policy 71 - Student Discipline if a ground for an appeal can be established.Read Policy 72 - Student Appeals, http://www.adm.uwaterloo.ca/infosec/Policies/policy72.htm

Note for students with disabilities

The AccessAbility Services (AS) Office, located in Needles Hall, Room 1132, collaborates with all academic departmentsto arrange appropriate accommodations for students with disabilities without compromising the academic integrity ofthe curriculum. If you require academic accommodations to lessen the impact of your disability, please register with theAS Office at the beginning of each academic term.

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