## Math 121      Linear Algebra and Applications

 Harvard Fall 2006

 Instructor: Thomas Lam

Course Assistant: Jakub Scholtz (jscholtz@fas)

 Lectures: Monday, Wednesday, Friday 12-1      Science Center 507
 Office Hours: Monday 3:15-4:30, Thursday 11:00-12:00. Science Center 435

(01/17) 5pm: Before calculating the final grades, I will likely add 6 points to everyone's final exam score. This should counteract the slightly unfair point system used for the TF question.
(01/17) 5pm: The question on the exam which no student answered completely was Problem 3(b) asking for square matrices satisfying AB-BA = I. This question was worth 3 marks, and 2 of them were awarded for noticing that tr(AB-BA) = tr(AB)-tr(BA) = 0. The last point was awarded for noticing that tr(I) can actually be 0, when the field has positive characteristic. In fact there are solutions to this equation over a finite field. For example, you may want to look for 2x2 matrices with coefficients in the finite field with 2 elements.
(01/17) 2pm: The average is 65. Slightly surprisingly the TF question was one of the most poorly done questions, with an average below 3.5. Yikes!
(01/17) 2am Yay! I'm done! You can email me to ask me for your exam score. The average for the final is probably lower than the midterms so I may be doing a bit of scaling...
In any case I will calculate some statistics.
(01/16) 10:30pm Whoa! High score so far: 97!
(01/16) 9pm: Nearly a third of the way through the exams. As expected, raw scores are not as high as for the midterms so far, but many of you have managed to struggle through some hard problems.m

Course description:
Real and complex vector spaces, dual spaces, linear transformations and Jordan normal forms. Inner product spaces. Applications to differential equations, classical mechanics, and optimization theory. Emphasizes learning to understand and write proofs.

Prerequisites: Mathematics 21a,b or equivalent.

Textbook: Linear Algebra, by Friedberg, Insel and Spence, Fourth Edition.

Exams: There will be two midterm exams given in class (dates Wednesday October 25, Monday November 20). There will be a final exam.

Grading: Problem sets (30%), MidTerm 1 (15%), MidTerm 2 (15%), Final Exam (40%).

Syllabus: We will cover most of Friedberg, Insel and Spence, omitting the starred sections in the later chapters. A detailed syllabus will be announced before each exam.

Homework:
Pset 1 (Due 09/27): 1.1 Q2(a) Q3(a), 1.2 Q8 Q16 Q19, 1.3 Q8(c,e) Q13 Q16 Q23 Q28, 1.4 Q12 Q15 Solution

Pset 2 (Due 10/04): 1.5 Q4 Q13[see p.555 for the definition of characteristic] Q17, 1.6 Q3 (d-e) Q14 Q17 Q22 Q26 Q31 Q32, 1.7 Q3 Solution

Pset 3 (Due 10/11): 1.3 Q31, 1.6 Q33 Q35, 2.1 Q5 Q11 Q12 Q18 Q21 Q35, 2.2 Q3 Q10 Solution

New rule for Psets: Psets will now contain a list of problems which you hand in, and a separate list which is "recommended".
Pset 4 (Due 10/18): 2.1 Q26, 2.2 Q12, 2.3 Q12 Q13 Q21, 2.4 Q3 Q5 Q17 Q22 Solution
Pset 4 (Recommended): 2.1 Q25, 2.2 Q9, 2.3 Q3 Q11, 2.5 Q3(d)

Pset 4.5 (Due 10/23): 2.5 Q8 Q10, 2.6 Q4 Q12, 2.7 Q13 Q18 Solution

Pset 5 (Due 11/03): 2.5 Q7, 2.6 Q13 Q14 Q15 Q16, 2.7 Q12, 3.2 Q5(d) Q6(b) Q14 Solution
Pset 5 (Recommended): 2.6 Q6 Q10, 2.7 Q4(b), 3.1 Q2 Q4, 3.3 Q3(b) Q6

Pset 6 (Due 11/10): 5.1 Q1 Q3(b) Q4(e) Q7 Q14 Q17, 5.2 Q7 Q13 Solution
Pset 6 (Recommended): 4.4 Q4(a)(g) Q6, 5.1 Q6

Pset 7 (Due 11/17): 5.2 Q12 Q14(a) Q17, 5.4 Q3 Q25(a) Q33, 5.3 Q6 Q7 Solution
Pset 7 (Recommended): 5.2 Q18 Q20, 5.4 Q2(c) Q6(c) Q9(c) Q36, 5.3 Q2(e)

Pset 8 (Due 12/01): 7.1 Q2(a) Q3(a) Q7, 6.1 Q4 Q8 Q11 Q13 Solution

Pset 9 (Due 12/08): 6.1 Q20 Q24(d) Q25, 6.2 Q2(i) Q13 Q14, 6.3 Q9 Q12 Solution
Pset 9 (Recommended): 6.1 Q21, 6.2 Q2(a), 6.3 Q1 Q3(c)

Pset 10 (Due 12/15): 6.3 Q18 Q22(c) Q24, 6.4 Q8 Q17, 6.5 Q15, 6.6 Q2 Solution
Pset 10 (Recommended): 6.6 Q5 Q7 Q10, 6.7 Q3(a) Q6(a), 6.8 Q4(a)(b) Q7

Announcements:
(01/14) Here is MidTerm 1 and MidTerm 2
(01/06) More solution sets are up, and the solution to Midterm 2 is here.
• The Final is to be held at 2:15pm Tuesday 01/16 in Science Center E.
• The exam will have roughly 10 questions with each question worth 5-20 points giving a total of 100 points.
• The first question will be a True-False question.
• There will be significantly more proofs in the Final than in MidTerm 2. I expect to award slightly more points for proof-style arguments than for explicit numerical computations.
• You should expect the proofs to be at least as hard as the question on Midterm 1 and the hardest proofs on the Final to be at least as hard as the proof question in MidTerm 2.
• A raw score of 87 will guarantee you at least an A-.
(12/21) Syllabus for Final Exam: For Chapters 1-5, the material covered will be identical to the syllabi for Midterms 1 and 2 (see below). Here is what you have to know for Chapters 6 and 7:
6.1-6.4, 6.6: Everything
6.5: Everything before the section on "Rigid Motions" (p.385)
6.7: Everything apart from the sections "Polar Decomposition of a Square Matrix"and "Pseudoinverse and Systems of Linear Equations". You SHOULD study the sections called "Pseudoinverse" and "Pseudoinverse of a Matrix".
6.8: Everything before the section on "The Second Derivative Test..."
7.1: You will need to be familiar with the statement (but not proof) of the Jordan Canonical Form Theorem. You should know the definitions of generalized eigeigenvectors and generalized eigenspaces. You could be asked to find the Jordan Canonical Form of an explicit matrix.

(12/06) Final Exam Resurrection Policy Announced: There are two resurrection methods:
1. If you score more than a particular cutoff (tba~90% raw) on the final you will get at least an A-.
2. If you don't, your course grade will be calculated to be the maximum of either (a) 30% Pset 30% MidTerms 40% Finals, or (b) 30% Pset 66.6% Finals.

(12/05) I won't be able to make it to office hours on Thursday 12/07. Please email me for an appointment.
(11/15) More information for Exam 2: The exam will have two questions. The first problem involves computations involving a particular matrix. You will be asked to compute eigenvalues, eigenvectors amongst other things. The second problem will ask you to prove something. It is worth much less than the first problem.
There are a total of 110 points on the exam, and 100 points is a full score.

(11/8) Syllabus for Exam 2 (November 20) The exam will focus on topics covered after the first midterm, and spend most of the time on Chapter 5. Here is what you have to know:
Chapter 3: 3.1-3.2. The only other thing you need to know is to be able to solve a system of linear equations if I give you one.
Chapter 4: 4.4. You do not need to be able to prove any of these facts.
Chapter 5: the whole chapter, excluding the section called "Invariant subspaces and directed sums" in 5.4. You CAN be asked about applications inclduing Markov chains and/or systems of differential equations.

(10/30)New Date (November 20) announced for MidTerm 2! E-mail me asap if you have a conflict. Also there is a new look to the webpage! There are now horizontal lines.

Solution for Midterm 1