## Math 102: Linear Algebra, Fall 2010

under construction!

Office hours: MW: 3:30-4:30 and by appointment (just talk to me after class, or email me)

Office: APM 5256, tel. (858) 534-2734

Teaching assistants: Jeremy Greene (email: j1greene@math.ucsd.edu) office hours: MW10-11:30 APM 6434 and Michael Kelly (email: mbkelly@math.ucsd.edu) office hours: F10-12 APM 6333

Computation of grade: The grade is computed from your scores in the final (50%), 2 midterm2 (20% each), and homework (10%). Passing grade for final required for passing the course! I will make a practice final available before the real final.

Dates of exams:

Midterms: 10/20 and 11/17 in class

### Texts

• Gilbert Strang: Linear algebra and its applications, Thomson, fourth edition.

Syllabus: This is a second course in linear algebra focusing on computational aspects and applications, yet presenting the geometric concepts. We start with a rapid review of the basic methods to solve systems of linear equations and the associated geometric subspaces and concepts. The applications will cover graphs and networks, least square problems, fast fourier transform, difference and differential equations and numerical solutions of these. The course will go further than a first course in factorizing matrices. Diagonalization produce factorizations of most square matrices but in general we have triangularization and the Jordan normal form. Gaussian elimination and Gram-Schmidt orthogonalization produce factorizations but a more useful one is the Singular Value Decomposition, which in particular can be used to construct a Pseudoinverse when there is no inverse to solve least square problems.

Tentative detailed syllabus (may change, i.e. we may go a bit faster or slower than indicated):

Week 1 (until 10/1): 1.1-7, 2.1 Matrices and Gaussian elimination functions, vector spaces and subspaces
Week 2: 2.2-5 solving Ax=b, linear independence, basis and dimension
Week 3: 2.4-6 The four fundamental subspaces, graphs and networks, linear transformations
Week 4: 3.1-3.4 Orthogonal vectors and subspaces, projections, least squares, orthogonal matrices, Gram-Schmidt,
Week 5: 3.5: Fast Fourier Transform, 4.1-4 Determinants
Week 6:
Week 7:
Week 8:
Week 9:
Week 10:

Homework assignments

Homeworks need to be turned in on or before the stated date, usually a Wednesday at or before 5pm. A mailbox on the 6th floor of APM should be available, but check with the TAs first in the first week. No homework needs to be turned in on Wednesdays when a midterm is scheduled. However, some of the homework may also be part of the material being asked for the midterm. It is very important that you do the homework problems as most of the exam problems will be variations of homework problems.

Disclaimer: I will try to get the homework assignment on the net in time. Due to time and other limitations, this may not always be possible. The fact that there is no assignment posted for a particular date does therefore NOT necessarily mean that no homework is due.

for 9/29: Sec. 1.2: 3, 10, Sec. 1.3: 3 (misprint: equation 2 --> equation 1, equation 3 --> equation 2), 18, 31, Sec. 1.4: 6, 10, 30. 32

for 10/6: Sec. 1.5: 1, 4, 15, Sec. 1.6:4,6,22,35,50, Sec. 1.7: 3,6, Sec. 2.1: 3, 7abcf, 8, 25, 26,

for 10/13: Sec. 2.2: 5, 8, 10, 24, 25, Sec: 2.3:2,10, 12,13,20,26,30, Sec. 2.4: 2,5,8,27,28,

for 10/27: Sec 2.5: 6, 8, Sec. 2.6: 6, 7, 8, 9, 16, 18, 22, 33, Sec. 3.1: 2, 7, 11, 14, 16, 19, 22, 32, 37, 44, 51, Sec. 3.2: 14, 17, 19, 21,

for 11/3: Sec. 3.3: 4,6,12, 17, 22, 27, Sec. 3.4: 13, 15, 16, 21, 23, (30 removed), Sec. 3.5: 11,14,

for 11/10: Sec. 4.2: 2,7,10,12,14,18,28, Sec. 4.3:3,5,28,43, Sec. 4.4: 5, 10, 14, 18,

for 11/17: need not be turned in, but relevant for midterm. Solutions will be posted below: Sec. 5.1: 5, 7, 14, 25, 27, Sec. 5.2: 4, 5, 7, 8, 15, 21, 29, 30, 34, 40,

for 11/24: Sec. 5.3: 2, 8, 10, 12, 15, 25, 28, Sec. 5.4: 1, 2, 3, 5, 8, 9,

for 12/1: Sec. 5.5: 16,17,18,36,38, 41,44,46, Sec. 5.6: 3, 8, 11, 13, 17(use 5R on p 296), 25, 31, 41, 44,

for final (need not be turned in, but relevant for final; solutions will be posted later): Sec 6.2: 2,4,8,23,27,29,30, Sec 6.3:2,3,5,10,15,19,

Solutions for midterms: The TAs did go over the midterm in sections. So we will not post solutions for the midterms. However, I will indicate below how the problems of the second midterm are similar to homework problems, for which you can look up the solutions, or I give some other indications how to do them.

Problem 1(a) was like Problem 7 of Section 2.6, but easier, and for (b) you only had to square the matrix.

Problem 2(a) was Gram-Schmidt (solution: (1,2,2,0) and (0,2,-2,1)), Problem 2(b) was like Problem 16 in Section 3.1; for instance, you can solve it by calculating the null space of the 2 by 4 matrix with rows (1,2,2,0) and (1,4,0,1). Problem 2(c) is just the projection u of x onto S, which is u = (1,10/3, 2/3, 2/3)^T, and for 2(d) we have u as in 2(c) and v = x-u.

In Problem 3 you calculate the determinant by putting it into echelon form (solution: 1), and for (b) det(2C)=8 det(C).

Problem 4(a),(b) was like Problem 14 of Section 5.1. To review: B has rank 1, and hence the null space has dimension 3, and we have had several problems where one calculates a basis for a nullspace. For a rank 1 matrix, any column vector is an eigenvector. In our case, B can be written as B=vv^T, where v^T=(1,-1,1,-1). Then you see that Bv=vv^Tv=4v. For 4(c) you just have to know that the columns of S consist of a basis of eigenvectors of B, which have been calculated in parts (a) and (b), and that the diagonal entries of Lambda are the eigenvalues of B, i.e. 4,0,0,0 to solve part (d) (here we assume that the first column of S is the eigenvector for 4, and the following three column vectors are a basis for the null space of B).

Final: We will have the same rules for the final as for the midterm. One cheat sheet, no calculators, books or other tools. Please bring bluebook/paper. I will post solutions for homework problems below. The new problems start with posting 8, part of which was already made available for the second midterm. Here is also a practice final given by another professor, with solutions. Please read below how my final may differ from that practice final, and for further tips.

Office hours for exam week: Jeremy: TW 9-12 (APM 6434), Hans Wenzl: MT 3-4+ (i.e. I'll stay beyond 4 if there are students around) (APM 5256)

More remarks: The practice final has two problems concerning calculating determinants, and two problems concerning solving linear equations and fundamental subspaces. Probably, our final will contain somewhat fewer problems of that type. Also, we have not covered material for question 8(c). Instead, some of the following problems may be on the exam:

- Calculate SVD for a given matrix

- Matrix of a linear transformation

- Exponential or large power of a matrix; solution of system of differential equation

- Properties of positive definite matrices, of symmetric matrices, Hermitian matrices

- Gram-Schmidt, projections

Midterm: The second midterm takes place in class on Wednesday, 11/17. The material goes primarily over the assignments for 10/27 until 11/17. Previous material will only be relevant if it is needed in connection with problems of these later sections. You are allowed to use one hand-written cheat sheet, but no books or calculators. Below is a practice midterm with solutions (but only look at the solutions after you have tried the problems in serious. You can also find solutions for homework problems below that.

These are practice midterms for the first midterm. You could now also have a look at problems 5 and 9 of the first practice midterm.

Here is another practice midterm

Homework Solutions: Here are solutions of homework problems. The ordering may sometimes be a bit different, i.e. you may find the solution in not exactly the assignment you were looking in. If you can not find it, check the other assignments.

------------------------------------------------------------------------------

Assignments for a previous course. These should give you a pretty good idea what to expect; the actual assignments will be quite similar or possibly even the same. But the relevant assignments will be above the dotted line

for 4/4: Sec. 1.2: 3, 10, Sec. 1.3: 3 (misprint: Eq. 2 --> Eq. 1, Eq. 3 --> Eq. 2), 18, 31, Sec. 1.4: 6, 10, 30. 32

for 4/11: Sec. 1.5: 1, 4, 15, Sec. 1.6:4,6,22,35,50, Sec. 1.7: 3,6, Sec. 2.1: 3, 7abcf, 8, 25, 26, Sec. 2.2: 5, 8, 10.

for 4/18: Sec. 2.2: 24, 25, Sec: 2.3:2,10, 12,13,20,26,30, Sec. 2.4: 2,5,8,27,28, Sec 2.5: 6, 8.

for 4/25: Sec. 2.6: 6, 7, 8, 9, 16, 18, 22, 33, Sec. 3.1: 2, 7, 11, 14, 16, 19, 22, 32, 37, 44, 51, Sec. 3.2: 14, 17, 19, 21,

for 5/2: Sec. 3.3: 4,6,12, 17, 22, 27, Sec. 3.4: 13, 15, 16, 21, 23, 30, Sec. 3.5: 11,14,

for 5/9: Sec. 4.2: 2,7,10,12,14,18,25,28,31, Sec. 4.3:3,5,28,34,36,43, for review see links to practice exams below

for 5/16: Sec. 4.4: 5, 10, 14, 18, Sec. 5.1: 5, 7, 14, 25, 27, Sec. 5.2: 4, 5, 7, 8, 15, 21, 29, 30, 34, 40,

for 5/23: Sec. 5.3: 2, 8, 10, 12, 15, 25, 28, Sec. 5.4: 1, 2, 3, 5, 8, 9,

for 5/30: Sec. 5.5: 16,17,18,36,38, 41,42,44,46, Sec. 5.6: 3, 8, 11, 13, 17, 25, 30, 31, 41, 44,

for 6/6: App B:1,5,6, Sec 6.2: 2,4,8,23,27,29,30,34, Sec 6.3:2,3,5,10,15,19, homework need NOT be turned in on Friday - but the material is relevant for the final! NO CHANGE OF ALLISON'S OFFICE HOURS THIS WEEK, office hours for exam week posted below.

Midterm The material for the midterm will go over the first 6 homework assignments. You are allowed one hand-written cheat sheet, but no other notes, books or calculators. You can find some practice exams below. You would only have to do problems from the first two exams for each term; you need not worry about problems dealing with material we have not covered yet.