W  8/19       Explanation of course policies/ basic concepts of linear algebra:  geometric vectors, R^n, vector spaces

F    8/21       Vector spaces: examples, formal properties/ Subspaces: examples

M  8/24        Examples of subspaces (intersection, sum of subspaces, subspace spanned by a set)/ Linear transformations: basic examples

W  8/26        Matrices acting on vectors/ linear systems- concrete and abstract formulations; range and column space
                     HOMEWORK SET 1
                      HW1- solutions
                     10 problems; due date Friday, Sept. 4. Bring a printout to class on Friday 8/28 (for discussion).

F    8/28      Nullspace and uniqueness/row space/ discussion of Hw set 1

M   8/31      Linear independence, basis, dimension

W   9/2        Simple computational examples/ bases and defining equations for a subspace/ orthogonal direct sum, orthogonal complement
                    Worksheet-computational examples
                    (Examples similar to those seen in class- not due as homework, but important for practice.)

F    9/4       HW set 1 collected (at the beginning)/ Dimension theorems/ transpose matrix

M   9/7       No  class (Labor Day)

W   9/9       Geometry of general linear systems/ linear isomorphisms, rank
                    The geometry of linear systems
                    (PDF handout, 6 pages; corrected and expanded version, 9/9/09). This handout summarizes the core information for
                     this part of the course.  All subsequent computational problems
                    (and much of the rest of the course)  will based on this theory.
                    It is important to think about this material, understand it and `internalize' it. If you spot a mistake, or if anything is unclear,                  
                    please let me know.

F    9/11      The row reduction algorithm/ row equivalence/row-echelon form
                     row-reduction summary  (includes some proofs of facts mentioned in class, and two questions to think about.) final version, 9/18
                    HOMEWORK SET 2  (due Friday, Sept 18, at the start of class)

M  9/14      Examples involving row reduction

W  9/16      More examples involving row reduction

F   9/18      composition of maps and matrix product/computation of the inverse by row reduction

M  9/21      upper triangular, symmetric and skew-symmetric matrices; MATLAB examples (time permitting)

W  9/23      Question-answer session (if there are questions) Bring printouts of the online handouts and homework.
HW set 2 and the section "matrices of maximal rank" (handout of 9/9) will be discussed.

F   9/25      Exam 1 (included: lectures up to 9/21, online handouts)
                   Practice test: see web page for my Math 251/fall 2005 (exam 1)
                   In Cullen's book, see: 1.3,1.4,1.5,1.7,2.1,2.2,2.3 (however, your lecture notes are more important, since the
                    exam will be based on the lectures, not on this reference).
                    20 "practice problems" from Cullen's book [given here by request]  1.3: 9,10,12/1.5: 4/1.7: 1,2,8/1.8: 2,3/
                    2.1: 8,10/2.2: 4,6/2.3: 1,2,4/ 2.4: 2,3,6,7
                   Exam 1

M 9/28        change of basis formula for vectors (and subspaces)/ brief discussion of Exam 1

W 9/30        determinants
                    Notes on determinants
                     (final version, 10/2)

F  10/2        change of basis formula for linear transformations
                     HOMEWORK SET 3
                     (due  10/9; will be discussed in class on Monday 10/4)
10/5-10/9       adapted bases/projection operators/similarity/eigenvalues
                       Linear transformations defined geometrically
                      (This handout, written in 2005, describes the main points in this part of the course. Reading it in
                       parallel with the lectures will help you understand them. Let me know if you have questions.)

M 10/5        change of basis in domain and range/ rectangular matrices in an adapted basis/projections

W 10/7       similarity of matrices:/orthogonal  projections/expansions, eigenvalue, eigenspace/projections in general

F 10/9         Orthogonal projection and reflection on a hyperplane: geometric formulas (based on the unit normal)
                  matrix groups: general linear group, invertible upper triangular

M 10/12     Equivalence relations between matrices: invariants, standard forms/ the trace
                  The trace-determinant formula (for matrix-valued differentiable functions)
                    Equivalence relations for matrices  (final version, 10/12)
                    HOMEWORK SET 4  (due date: 10/21)

W 10/14      orthonormal bases/Gram-Schmidt/application to orthogonal projections
                    orthonormal bases and the orthogonal group
                    solutions to hw3
F 10/16        FALL BREAK (no classes)

M 10/19        orthogonal group/rotations in R2 and R3/OT decomposition
                      Orthogonal matrices, rotations and reflections (2005 handout)

W 10/21        HW 4 due/ Q&A session
                      solutions to hw4
                       correction (problem 5)

F 10/23         Exam 2 (material from 9/28 to 10/19)
                      Exam 2 (problems)
                       Exam 2 (solutions)

M  10/26     Least-squares approximate solutions (application of projections)/associated `normal system'

W  10/28     Least-squares fitting to data: linear regression, residuals add to zero, approximation by
                    higher-degree curves (quadratic example)/ Brief discussion of exam 2
                    A:nnouncement: Optional exam on Wed 11/4 at 6PM: material- same as exam 1+exam 2(combined)
                                                           (grade will replace the lowest of the two grades)
                    Room: HSS 204

F  10/30      Characteristic polynomial, eigenvalues/diagonalizable matrices/eigenspaces for 2X2 matrices

M   11/2     Complex eigenvalues of real matrices/ interpretation in terms of R^n
                    complex eigenvalues of real matrices
                    (2005 handout)

W   11/4     Analysis of linear operators: Complex-diagonalizable and non-diagonalizable matrices/ standard forms for n=2/
                    invariant subspaces
                   Similarity classification of matrices (n=2,3)
                    (2005 handout)
                    Optional test
                    optional test-solutions
                    (note: the order of the problems in the solutions  differs from that on the test, but it is easy to figure out the correspondence.)

F    11/6    HW 5 posted (due 11/13)
                   HOMEWORK SET 5
                  Principal axis theorem (preparatory material)
                   The Principal Axes Theorem
                   (includes: quadratic forms, singular value decomposition)

M    11/9  Principal Axes Theorem

W    11/11 Quadratic forms

F     11/13  Singular Value Decomposition (start)

M    11/16  Singular value decomposition (end)-
                  HOMEWORK SET 6: the 10 problems are found in the handout "Principal Axes Theorem" (posted 11/6)- DUE 11/23

W    11/18  Application: solution of first and second order recursions- the Fibonacci sequence
                   Solution of linear recursions (includes  Fibonacci example and 4 problems)
F      11/20  Standard forms for n=3 (see handout posted 11/4)
                   Practice problems on matrix powers (with solutions)
                    Graded HW5 returned

M    11/23 Discussion of Hw sets 5 and 6/Hw6 due

W   11/25:  Exam 3 (material included: lectures from 10/26 to 11/16)

F     11/27   Black Friday (no classes)

M     11/30     Discussion of problems from 11/18, 11/20 lectures
                       PRACTICE PROBLEMS: see handouts posted 11/4 (n=3) and 11/20/ course evaluations

FINAL EXAM:  Tuesday, Dec.8, 2:45-4:45 (section 5)/ Wednesday, Dec 9, 10:15-12:15 (section 3)-in the usual classroom
Fall 2009 Final Exam Calendar (Registrar's Office)

Material included: comprehensive. The test will consist of 8 problems:, including at least one on the material
discussed in lecture on 11/18,11/20,11/30.

STUDY GUIDE- you should at least review the following problems (try to do them without looking at the solutions)
Exams 1, 2, 3  and optional test
Homework sets 1 through 6
For the material introduced after Exam 3: problems in the handouts (i) similarity classification of matrices for n=2,3 (11/4)
(ii) Solution of linear recursions (11/18) (iii) Practice problems on matrix powers (11/20). (i) and (iii) include answers.
For additional practice problems:  Fall 2005 course (here)-exams, including final.

OFFICE HOURS prior to the final:  Tuesday 12/8, 9AM-12PM (section 5 students)/ 5PM-8PM (section 3 students)
Remark: I'll be out of town from Wednesday 12/2 to Monday 12/7, but will answer questions by e-mail.

Final Exam