MATHEMATICS 251, LINEAR ALGEBRA-FALL 2010-COURSE LOG

Th 8/19       Course policies/  linear systems in 2 and 3 variables, matrix form, examples of elimination and parametrization/
                    matrix times column vector (def), matrices define linear maps
                    Problems (1.1):  6, 10, 15, true-false (a) to (h)

Tu 8/24     Row-echelon form, rref, row reduction/ definition of nullspace and range/ basis and parametrization of the nullspace/
                 finding the condition of solvability and the general solution of a linear system
                  Problems (1.2) 3,19,21,25,27,37,41/true-false (a) to (i)
                  Problems  (1.6) 15,17

Th 8/26: matrix products/invertible matrices/computation of the inverse (1.3, 1.4 and 1.5)
              section (1.6): reading assignment
                Problems: (1.3) 27, 29 (1.4) 18, 28, 39, 54, 55 (1.5) 13,15,19

Course project- details
This assignment consists of writing a short paper (4-5 pages) describing an  application of material found in the text
(whether the material is due to be covered in lecture or not) to a problem in an area of science or engineering
of interest to the student; typically this is found in a text for a more advanced class in the student's major.
Rules: 1- The application in question may not be one already described in the text (or in a different linear algebra text).
For examples of what is meant by "describing an application", see sections 1.9, 9.3, 9.6 in the text.
2- Sources: written sources must be textbooks or original papers in the area of application. Internet sources are excluded.
3-Plagiarism (copying large chunks of material without attribution) will be detected and punished (zero on the assignment, treated
as academic dishonesty).
4-Human sources: by all means, ask a more advanced student or a professor in your major department (or intended major) for help
with this. Part of the goal of the assignment is to have them help you understand why this course is required. The names of people
who provided assistance must be included in the paper.
5-Structure- the paper should  have the following sections:
                   1-Introduction: statement of the general problem and of its importance in the applied area;
                   2- Formulation of a specific instance of the problem (numerical example)
                   3- Solution of the problem in part (2), based on material included in the text for this course
                   4- Conclusion (interpretation of the result, in terms of the science/engineering application considered).
                   5-References (texts/papers consulted, people providing assistance.)
6-Due dates: the project is due in two steps. On Oct. 14  you  must turn in a draft, including at least the first section (description
of the problem). I will review this and make comments. If this draft is not turned in on this date-no need to go on (zero on the assignment).
The final paper is due on the last day of class, Nov.30.

Tu 8/31  linear systems and invertible matrices (nullspace, range)/ transpose and symmetric matrices/triangular matrices
              Problems: (1.6): 18, 21, true/false (a) to (g)
                               (1.7): 20, 21,22,32,33,41, true/false (all)

Th 9/2  some applications of linear systems (section 1.8)
             Problems (1.8): 3, 7, 11,15

Tu  9/7: Chapter 2 (determinants)
             Problems: (2.1): 36,38,39,40,41 (2.2): 28, 29, 34, 35, 36 (2.3): 33, 34, 38, 39 Supplementary: 29, 33, 35, 36

Th  9/9 Geometric vectors and analytic geometry
              Problems (3.1) 28, 30, 33 (3.2) 15, 16, 17, 18, 19, 32, 33, 34 (3.3) 5,7,9,19, 25

Tu  9/14  Orthogonal projections, orthogonal complement / parametric and non-parametric forms of lines and planes/row space and nullspace:
orthogonality. Problems  (3.4) 9,13,17,21,22,23,24

Th 9/16  Column space: interpretation in terms of linear systems, consistency conditions. Finding bases for row space, nullspace, column space.
(examples in row-echelon form, and for a general matrix via row reduction). Relationship with the general solution of Ax=b.
Problems (4.7) 3,4,5,6,7,8,11

Tu 9/21  Subspaces of R^n: definition, examples. Subspace spanned by a set, or defined by a system of equations. Problem 1: from spanning set to
a basis. Problem 2:  from a system of defining equations to independent defining equations. Problem 3: from defining equations to a basis (parametrization
of the nullspace). Problem 4: from a basis to defining equations (=basis for the orthogonal complement) Definition and properties of the orthogonal
complement of a subspace. This material is included in the Thursday test.
Practice problems on this material


The first test will be on Thursday, 9/23
Sections included: sections 1.1 to 1.8, 2.1 to 2.3, 3.1 to 3.4, 4.7
Problem session on Tuesday 9/21: HBB 103, 5:45-7:00
Office hours: Tuesday 5:00-5:45, Thursdays 12:30-1:30 (or by appointment)

Exam 1 (with solutions)

Tu 9/28  Axioms for vector spaces/ subspaces, examples/linear independence/basis for a subspace, dimension/coordinates of a vector in a basis.
Problems (4.2) 1,2,3,4,5 (4.3) 3,7,8 (4.4) 3,9 (4.5) 7,8,9,13,16

Th 9/30 Concept of linear transformation/change of basis formula
Problems (4.6) 1,2,12(b)(e),13(b)(e)/ (4.9) 10, 11, 16, 17

Tu 10/5 linear transfs. defined geometrically I: matrix free expression for orth. projection and reflection on subspaces of dimension n-1/
change of basis formula for linear transformations/application: matrices for oblique projections in R^2 and R^3 (remark: this material is dispersed
throughout several sections in the text; your class notes and the handout below have precedence here.)
Problems (4.9): 16,17,18,19
Reading assignment: before the next lecture, read the handout below.
linear transformations defined geometrically

Tu 10/12 Linear transformations defined geometrically II: rotations in R^2, expansions/contractions, eigenvalues and eigenspaces (based on handout above)

Th 10/14 PROJECT DUE (1st. draft) Rank, nullity, fundamental spaces (4.8)/ orthonormal bases (6.3)
Problems (4.8):2,4,5,7,8,12,16 (6.3)9,17,26

Tu 10/19 Gram-Schmidt (6.3), Orthogonal matrices, reflections in R^3, projections using an o.n.basis (handout)
orthogonal matrices
Problems: (6.3)21,23 (7.1) 1,3,4 (4.7) 15, 16, 20

Th 10/21 Rotations in R^3, Eigenvalues, diagonalization (5.1, 5.2, 5.3) algebraic and geometric multiplicities of eigenvalues
Problems: (5.1) 5,8,13,15,23 (5.2) 7,9,15,17,23

Tu 10/26 Complex eigenvalues. [REVIEW COMPLEX NUMBERS before 10/26 class-Appendix B]
PROBLEM SESSION, HBB 103, 5:45-7:00
Problems: (5.3) 15,17,19,21,23,25

Th 10/28 EXAM 2 Included: lectures and handouts from 9/28 to 10/26
Exam 2
solutions

Tu 11/2 Discussion of Exam 2/ least-square solutions of inconsistent systems (6.4)
Problems (6.4): 3, 5, 9, 10, 15

Th 11/4  Least-squares fitting to data (6.5)/ orthogonal diagonalization (7.2)
Problems (6.5) 1, 3, 11 (7.2) 1,3,5,14,17

Tu 11/9   and Th 11/11: Quadratic forms: diagonalization, geometry of level sets, optimization
Problems  (7.3) 5,7,11,13,15,25,32 (7.4) 1,3,5,11,13,15,17

Tu 11/16 and Th 11/18: application to systems of equations differential; stochastic matrices and Markov chains
Problems (5.4) 1,2,3,4 (4.12) 15,16,17,18  

Tu 11/23: EXAM 3 Included: lectures from 11/2 to 11/18
Exam 3
solutions

Th 11/25: Thanksgiving

Tu 11/30-solution of exam 3, course project due, course evaluations

FINAL EXAM: Tuesday, 12/7, 12:30-2:30
The final will consist of eight problems, taken from exams 1,2 and 3 (with slight changes)