## Math 315—Advanced Linear Algebra (Spring 2019)

Instructor: Ben McMillan
Email: bmcmillan@math.stonybrook.edu
Location: T/Th 2:30pm--3:50pm in Library N3063

### Course Information:

The course syllabus is here. Some critical points:
• The midterms will be in class on February 14 and April 2
• My office hours this term are Tuesdays 1–2pm and Fridays 11am–12pm in the Simons Center 508. I will also be available in the MLC (S-235) on Thursdays 10:30am–11:30pm.

### Schedule and Homework:

The following is a tentative schedule for the course. As homework is assigned it will be posted here. You are encouraged to work with others, but please make sure to write up solutions in your own words (this will help you on the exams and quizzes!).

Unless stated otherwise, the homework is from the corresponding section of the book.

All of the problems assigned in week n are due to your TA (at the start of recitation) in week n+1.

It will be expected that you have done the suggested reading.

Problems in square brackets are ones that I think you will benefit from, but won't be graded. You should still do them!

Week Date Topic(s) Covered Reading Homework
1 1/29Abstract Vector Spaces1A, 1B1A: 1, 15
1B: 1, 6
1/31Subspaces, Direct Sums1C3, 7, 8, 10, 24
2 2/5Spans, Polynomials2A1, 2, 6, 8, 11
2/7Linear Independence, Definition of Dimension2B2, 3, 5, 7, 8
3 2/12Bases2C1, 3, 8, 10, 11, 16, [17]
2/14Midterm 1.Review
4 2/19Linear Maps3A1, 3, 4, 8, 9 [do R--->R too!], 11
2/21Injectivity/Surjectivity & Kernel/Image3B2, 3, 7, 9, [11], 20, [21], [26], 28
5 2/26Linear Isomorphisms3D3B: 17, [18]
3D: 2, 4, 5, 8, 9
2/28Coordinates3C2, 3, 8, 14
6 3/5Equivalence Relations, quotient spaces3E12, 13, 16, 20
3/7Direct Products3E1, 6, [recall that a map of sets f:A--->B defines an equivalence relation ~ on A. Show that f is injective if and only if the quotient map A--->A/~ is an isomorphism.]
7 3/12Dual Spaces3.F7, 9, 13
3/14Tensor Products.here
8 3/19Spring Break..
3/21Spring Break..
9 3/26Practice with Tensor Products..
3/28Change of Bases..
10 4/2Midterm II.Review
4/4Trace.here
11 4/9Eigenvalues/Eigenvectors5A11, 12, 15, 21, 25
4/11Eigen-decomposition5B1, 2, 5, 10, 11
12 4/16Generalized Eigenvectors5C
8A
6, 12
1, 4, 5, [12], 15, [16, 17]
4/18Jordan Canonical Form8B
8D
1, 9
4, 5
13 4/23Inner Products6A8, 10, 14, 18, 23
4/25Orthonormal Bases6B1, 4, 5, 11, 12
14 4/30Riesz Representation Theorem6B
6C
7, 8, [9], [15]
[4], 5, 10
5/2Complexification9A[Read at least 276-278] 2, 3, 4, [17]
15 5/7Self Adjoint operators7A[1, 3, 6, 8 [can you find a natural complement?], 20]
5/9Representations, symmetrizers and the determinant..
16 5/7Study.Review
5/9Final Exam5:30-8:00pm.