Professor David Blecher

Welcome to Advanced Linear Algebra

Class time and place: M-Th (and some Fridays) 12:00PM - 2:00PM in Room SR 140 - 6/2/2008 to 7/3/2008

Office Hours Tu, W, Th 11-12 (or by appointment), in PGH 622. The grader (Georgios) will have office hours too, in PGH 605, MW 10:00-12:00.

My email address: dblecher@math.uh.edu and phone number (713)-743-3451. Fax (713)-743-3505

Textbook: If you are taking the second semester too (which I am not teaching), get "Linear Algebra", 2nd Edition, by K. Hoffman and R. Kunze (Publisher: Pearson/Prentice-Hall). However my lectures will be based on "Linear Algebra done right", 2nd Edition, by S. Axler (Springer), and this book is inexpensive, so get it (and you must get it if you are not taking the second semester).

Prerequisites: Math 2331 and a minimum of three semester hours of 3000-level mathematics

Final Exam 11am - 2pm, Thurs July 3

Description of the course: This class falls in the category of those math courses focusing on theorems and proof, requiring a lot of abstract logical reasoning. However amongst such classes, this is one of the easiest: the proofs and concepts are not difficult. This material is foundational for most branches of mathematics, and for related sciences and engineering.

Syllabus: Systems of linear equations and matrices, Vector spaces, linear independence, subspaces, direct sums, finite-dimensional spaces and bases and dimension. Linear operators and their matrices, null spaces and ranges, invertibility. Eigenvalues and Eigenvectors, polynomials with real and complex coefficients, polynomials of operators, diagonal and triangular matrices. Inner-product spaces, orthonormal bases, orthogonal projections, adjoints. Selfadjoint and normal operators, the spectral theorem, positive operators, the polar decomposition, square roots, generalized eigenvectors, the characteristic polynomial, Jordan form. Trace, determinant, change of basis, volume.

Course grade will be out of 500 points; consisting of 100 points for each of two semester tests, 100 points for homework/quizzes, 200 points for the final exam.

Students in this course should keep monitering this website, particularly just before tests (for example if there is aal exam. last minute change in a test date)

June 5: Last day to drop the class and not have it count towards the Enrollment Cap/to get a full refund? Last day to drop and get a W is June 20.

To do after Class 1, 2, 3: Read the long handout I gave out in class. Additional homework: Write down the proof that examples 5 and 8 from class 6/2, are vector spaces, including all details. Homework 1 (that is, ALL the checked items on the handout) is due Friday 6 June.

There is class on Friday June 6, June 13, and June 20, but not June 27. June 20 will be a test date.

Basically you main task right now is to carefully read your classnotes each day, making sure that you have mastered all the tiny details. Also memorize the definitions and basic facts (statements of theorems, propositions, etc). Also keep up to date with your homework.

We have finished CH 3, Linear transformations, and the homework is due Monday. The problems that will be graded were mentioned in class thursday. We have finished CH 4, and those homework questions can be handed in with HW CH 5. We have finished CH 5, and I gave out the HW for CH 5 on last Thursday. In addition for HW 5, find a basis such that M(T) is triangular where T is the operator on R^3 given by left multiplication by the matrix with rows [0 , 1, 0], [2, -2, 2], [2, -3, 2] ; also write down what is M(T). HW 4 and 5 is due Wednesday.

Test 1 is scheduled for June 23, a Monday. Friday 20 there will be no class, it is a preparation day for the test. If you have questions you can ask the TA then. in PGH 605, I will be out of town.

Proofs to memorize for Test 1: the list is: CH 1 Prop 11, CH2 Lemma 1, Prop 2, Theorem 4, Props 5, 6, 7. CH 3 Props 1, 2, 3, Cor's 1, 2, 3, 4, 5, Prop 7, CH 4 none, CH 5: Cor 1, Lemmas 1 and 2, Theorem 2, Prop 2, Cor 2 (the last corollary in CH 5).

To study for Test 1: Test 1 will cover up till and including Chapter 5. Read classnotes several times, assimilating and understanding everything. Memorize all definitions and statements of Facts, Theorems, Propositions, Corollaries, etc. Carefully review all homework, and their keys, learning from your mistakes. Understand how the concepts fit together. Memorize proofs from the list in the last paragraph, in the way I told you in class. The typical question on your test will look like this: Q 5 (a) What is the dimension of a finite dimensional vector space V? (b) Prove that in an n dimensional vector space V, any l.i. set of length n is a basis. (c) Show that if U is a subspace of V with the same dimension, then U = V. (d) Complete the sentence: If U, W are subspaces of V then dim(U + W) = ... .

The last date to drop the course is Friday June 20. Since Test 1 is on Monday, I would use your graded homework, and whether you have followed all the notes till now when reading them daily in full detail as I insisted, as a test for whether you should drop or not.

If you plan to drop, you can bring a drop form to the secretary of the math department on Thursday or Friday and she can sign for me

Apologies, the test was insanely long. It will of course be graded on a curve. So what matters is how well you did relative to the rest of the class.

Test 2 is on the last day of class (Tuesday). Instructions as for the last test. List so far of proofs to study: CH 6: the Cauchy-Schwarz and triangle inequalities (not their converses, ie not the case of these results where you have `='), Thm 1, Cor 3, Cor 4, Cor 7, CH 7 Prop 1, Cor 3, Thm 1, Cor 4, CH 8 Cor 1, Prop 4, Prop 5, Cor 2, Cor 3. CH 10 Cor 1, Cor 2.

As announced in class, the deadline for HW for CH 6 is Thu, for CH 7 and 8 is Friday. Please also note that the list of proofs to study above for test 2 has changed slightly. The test 2 will cover up to and including CH 10 Corollary 4.

All the notes up to and inluding Thurs 6/26 are up in the usual spot (...n4.pdf) Note I have changed the list of proofs to study yet again (removing anything about trace or det).

If you email me homework, do not send it as jpg, do pdf.

The test tuesday will be for the entire class period. The first hour will be as usual, and you will turn in your work after that. But in the second hour you will be able to tackle 4 or 5 questions that are `unseen problems' (as opposed to definitions, facts, theorems, proofs etc), with open books. These problems will be similar in nature to questions that appeared in the homework, but will not be questions straight from the homework. They will not be `very hard', and some of them may require thinking through principles from the class. Their point value also will be small compared to the rest of the test; so that the sum of their points will constitute probably around 1/4 of the total test grade.

Instructions for final exam coming. It is comprehensive, including CH 10. The instructions will be similar to the previous tests, however there will also be questions requiring use of Gauss (row) elimination, to find a basis etc, similarly to in the prerequisite course. List of proofs to study: From the test 1 material, add the proof that an operator on a f.d. complex vector space has an eigenvalue. On the Test 1 and 2 material, you can strike the proofs that appeared on the actual test 1 and 2. For CH 10, memorize the proofs of Cor 1, Cor 2, Cor 3, Theorem 2, Prop 4, Theorem 4 (det T = determinant(M(T))), and Corollary 6 (det(ST) = det(S) det(T)).

Test 2 key.

Note the total on the key is 100 (the points on the right of the page are correct, the ones on the left are not.

For the proofs on the final, a theorem whose proof in only one direction that appeared on Test 1 or 2, may be asked in the other directions in the final. The final is a bit more heavily weighted to the material in the last week or so of class.

I will be out of town today, so please ask any questions of the TA, Georgios Georgios.Kazasis@mail.uh.edu

More information, mock exams, etc will be added to constantly here below.