MATH 410-2 Matrix Analysis, Spring 2007
 
Lectures:

Text: Applied Linear Algebra, P. Oliver and Ch. Shakiban

Problem sets: Homework problems will be assigned on Wednesdays, and they are due the following Wednesday. There are many linear algebra packages available, and one can use them for solving some of the homework problems. However, you are encouraged to do the problems by hand. Solutions of the problems must include detailed explanations. Answers with no explanations will not be counted.

Exams: Two midsemester tests given during class periods on February 23 and April 9 plus a two-hour final. The final exam will take place on Wednesday, May 11, from 11:00am to 1:00 p.m.. Laptops and calculators will not be allowed during the exams.

Grading: The homeworks and the midsemester tests are worth 20 percent of your final grade each; the final exam is worth 40 percent of the final grade. Student must achieve the passing level on the final exam to pass the course.

Grades "W" and "I": The grade "W" will be awarded to any student who requests this grade before March 6. The grade of "I" can be awarded only in an exceptional case to a student who has a valid reason for not completing the course in time, who has not completed a small portion of the course, and who has shown a passing performance in class.

Announcements:

Problem sets:
1. Due on January 17: 1.1.1(c,e,f), 1.1.2, 1.1.3, 1.2.3, 1.2.4(c,e,f), 1.2.5(b,d), 1.2.7(a,c,e,g), 1.2.8(a,b), 1.2.11, 1.2.14, 1.2.20, 1.2.26, 1.2.32, 1.3.1(b,d), 1.3.2(d), 1.3.3(c), 1.3.5(b,c)
2. Due on January 24: 1.3.14(d,e), 1.3.15(b,d), 1.3.22(d,f,h), 1.4.1(b,d), 1.4.3(a,c), 1.4.10(b,d), 1.4.13, 1.4.19(b,d), 1.4.20(a,c), 1.5.1(b), 1.5.3(b,d,f), 1.5.4, 1.5.9(b), 1.5.13, 1.5.14, 1.5.24(e,g), 1.6.1(c,g), 1.6.2, 1.6.5, 1.6.17(c), 1.6.25(b,d)
3-4. Due on February 11: 1.8.2(c,g), 1.8.5, 1.8.7(c,e,h), 1.8.16, 1.8.17, 1.9.1(c,g), 2.1.6(b), 2.1.7, 7.1.1(a,c,e), 7.1.2(b,d,f), 7.1.7, 7.1.5(a, c, e, g, i)
2.2.2(all), 2.2.4, 2.2.6, 2.2.8, 2.2.14, 2.2.15(a,c,e), 2.2.18, 2.3.3(a,c), 2.3.4(b,d,f), 2.3.8(a), 2.3.9(b,d), 2.3.12, 2.3.21(a,c,e,g), 2.3.33(a,c,g), 2,4.1(b,c,e), 2.4.6(a), 2.4.8(a,c)
5. Due on February 23: 2.5.1(b,d), 2.5.2(b,d), 2.5.5(a,e), 2.5.6, 2.5.13, 2.5.17, 2.5.21(b), 2.5.22, 2.5.24(ii), 2.5.25(c), 2.5.26, 3.1.2(a,c,e), 3.1.4(a), 3.1.13, 3.1.19(b,d), 3.1.20(c), 3.1.21(b,d)
6. Due on March 7: 3.2.2(a), 3.2.6, 3.2.15(a), 3.2.19, 3.2.27, 3.2.36(a,b,c), 3.3.3(a,c), 3.3.6(a,c), 3.3.10(f), 3.3.11(b,e), 3.3.28(b,d,f), 3.3.29(b,d,f,h), 3.4.2, 3.4.22(ii, iv, vi), 3.4.27, 3.4.28
7. Due on March 14: 3.5.1(d,f), 3.5.2(d,f), 3.5.3(a,b,c), 3.5.7(b,d), 3.5.10(a-d),3.5.21(g), 3.6.28(a,e), 3.6.29(a-e), 3.6.30(c), 3.6.31, 3.6.32(c), 4.1.1, 4.1.2, 4.1.3(e)
8. Due on April 6: 4.2.3(a,c,g), 4.2.4 (a-c), 4.3.4(b,d), 4.3.5, 4.3.9, 4.3.14(c,e), 4.3.15(e), 4.4.1(c), 4.4.4, 4.4.8, 4.4.12(c,e), 4.4.13(c)
9. Due on April 13: 5.1.2(a,c), 5.1.4, 5.1.6, 5.1.27, 5.2.2(b), 5.2.5, 5.2.9(c), 5.3.1(b,d,e), 5.3.12, 5.3.27(b,d), 5.3.28(iii), 5.4.1(a,c)
10. Due on April 20: 5.5.2(d), 5.5.3, 5.5.7, 5.5.32, 5.6.1(c), 5.6.2(b), 5.6.3(b), 5.6.20(b), 5.6.24, 8.2.1(b,d,f,h), 8.2.6, 8.2.7(b), 8.2.17, 8.2.20, 8.3.2(b,d,f), 8.3.3(b,d,f)
11-12. Due on April 30: 8.3.6, 8.3.8, 8.3.15(b,d,j), 8.3.17, 8.3.21(b,d), 8.4.1(c,e), 8.4.14(b,d), 8.4.16(b,d), 8.4.35, 8.4.38(d), 8.6.1(b,f)
9.1.3, 9.1.11, 9.1.12(c), 9.1.16, 9.1.18, 9.1.26(b, f), 9.2.1(b,c,e), 9.2.3(b), 9.2.8, 9.3.3(b,d), 9.3.4(a,c) Practice test: Tasks set in PDF

Solutions to selected homework problems:
Problem set 1
Problem set 2
Problem set 3
Problem set 1 complete
Problem set 2 complete
Problem set 3 complete
Problem set 4 complete
Problem set 5 complete
Problem set 6 complete
Problem set 7 complete
Problem set 8 complete
Problem set 4
Problem set 5
Problem set 6
Problem set 7

Internet resources:
1. The following toolkit contains programs performing Gaussian elimination, LU-decomposition, matrix inversion, etc., and, most importantly, one can see how all the operations are performed on the step-by-step basis
2. Solving linear systems of big size is a challenging problem. You may want to look at world records from 1950's to 2002.
3. Linear algebra is used in many applications; image recognition is one of them. You can take a look at the face recognition demo.
4. The following interactive demo draws the least square line to a set of points.

APPROXIMATE SYLLABUS
MondayWednesdayFriday
January
 
10
Linear Systems. Matrices and vectors. Sections 1.1, 1.2
12
Gaussian elimination. Section 1.3
15
Martin Luther King Day
17
Pivoting and permutations. Section 1.4
19
Matrix inverses; transposes and symmetric matrices. Sections 1.5, 1.6
22
General linear systems. Section 1.8
24
Determinants. Section 1.9
26
Real vector spaces and linear functions. Sections 2.1, 7.1.
29
Real vector spaces and linear functions. Sections 2.1, 7.1
31
Subspaces; span. Sections 2.2,2.3		  
February
   
2
Linear independence and dependence. Bases. Dimension. Sections 2.3, 2.4.
5
Linear transformations. Section 7.2
7
The fundamental matrix subspaces. Seection 2.5
9
The fundamental matrix subspaces. Seection 2.5
12
Inner products. Section 3.1
14
Inequalities. Section 3.2
16
Review
19
First Test
21
Norms. Section 3.3
23
Positive definite matrices. Section 3.4
26
Positive definite matrices. Section 3.4
28
Completing the square. Section 3.5		  
March
   
2
Complex vector spaces. Section 3.6
5
Minimization problems. Sections 4.1, 4.2
7
Least squares. Section 4.3
9
Data fitting and interpolation. Section 4.4
Spring Break
19
Orthogonal bases. The Gram--Schmidt process. Sections 5.1, 5.2
21
Orthogonal matrices. Section 5.3
23
Orthogonal polynomials. Section 5.4
26
Orthogonal projections. Section 5.5
28
Orthogonal subspaces. Section 5.6
30
Review
April
2
Second Test
4
Dynamical systems. Section 8.1
6
Eigenvalues and eigenvectors. Section 8.2
9
Diagonalization. Section 8.3
11
Eigenvalues of Symmetric matrices. Section 8.4
13
Singular values. Section 8.5
16
Schur decomposition. Section 8.6
18
Linear dynamical systems. Sec. 9.1
20
Stability. Section 9.2
23
Two-dimensional systems. Section 9.3
25
Matrix exponential. Section 9.4
27
Matrix exponential. Section 9.4
30
Markov processes. Section 10.4		    
May
 
2
Review