MATH 251- FALL 2005- COURSE LOG

8/24     Outline of course; policies
solution of lin systems by elimination: 3X3 example
HW: 1.1: 4(c), 5(b); 1.2: 3

8/26      General linear systems; Gaussian elimination (1.1, 1.2)
A 4X5 example (1.2-10c); parametrizing the general solution (vector form)
HW (1.2) 5, 7a,c, 10b,  12 (in your head; justify), 13, 17

8/29       Vectors and matrices (1.3)
vectors in R^n: linear combinations, dot product- interpr of linear systems
matrix product: definition- special case: A(m x n) maps R^n to R^m
3 connections between matrix products and this map
HW (1.3): 7 c,d; 8 a; 14a, 16c

8/31         1.3(end); 1.4(inverses , 2X2 computations)
connections between products, systems, row operations
3 properties and 3 "non-properties" of the matrix product
HW(1.4) :7d, 8, 16
for 16: if the answer is "no" , justify by giving EXAMPLES  invertible A and B with A+B not invertible

9/2            QUIZ 1 (3 problems from HW up to 8/31, 15 min)
computation of inverses via row reduction (1.5)
inverse of partitioned matrices: problem 18 in 1.4
HW(1.5): 6c, 7c, 8b,c

9/5: LABOR DAY (no classes)

9/7           Invertibility and consistency of systems (1.6)
Theorem:  invertibility, consistency of systems and the rref  of a square matrix
Example: (i) determining consistency conditions for a 4X4 system via row reduction;
(ii) parametrizing the general solution when the system is consistent (iii) notion of
nullspace N(A) of a matrix and the form of the general solution.
HW: (1.6) 12, 15, 18, 21

9/9            QUIZ 2 (3 problems from HW for 1.5, 1.6)
General maps vs. linear maps: 1-1, and onto vs. existence/uniqueness of solutions. Range of
a linear map,  Ran (A)=column space
A  3X4 example,: conditions for consistency (=computation or Ran(A)),  general solution (=computation of N(A))
Where this is found in the text:   p.189/192,  p.247/250 (ignore "bases" for now)
practice problems:  p.198:  3, 4, 5a, 6a, 15 ; p.257:  2b, 3e, 4, 5b, c

9/12            3.5: lines and planes in 3-space: equations (inc. parametric eqns for planes), connection with systems
practice problems p.155:  3, 4, 5, 6, 7, 8, 9, 10, 11 (one item of each)

9/14:           REVIEW: emphasis on HW and practice problems above
lines, planes, and linear systems in 3 unknowns

9/16       FIRST EXAM: Friday, 9/16,  topics: material up to 9/12
NOTE: The exam will consist of 6-8 problems very similar to
the HW and practice problems listed above (or to examples discussed in class). So it should be clear how to study for it.
Exam 1 , with answers (PDF file)

9/19           subspaces of R^n- def, examples: range, nullspace, subspace spanned by a set of vectors
Rk. homog systems with no eqns < no. unknowns always have non-zero slns.
anti-examples: solution sets of non-homog lin syst, inequalities, nonlinear syst.
HW (section 5.2): 6, 7, 8, 11, 15 (turn in one item of each)
Problem 1: use the defn given in class- if subspace, explain why; if not, give example of vectors violating the defn

9/21             linear independence (of a set of vectors); basis, dimension (of a subspace)
HW(5.3): 3, 5, 7
HW(5.4): 8, 12, 16, 17
(Rk: in multi-item problems- if you turn in HW, only one item is enough, but you should be confident you can solve all items.)

9/23            discussion of HW
section 5.5: row space, column space, nullspace- computation of bases using row reduction (main theorem on how the spaces change)
Any subspace of R^n can be given either in parametric form (using a basis, with the number of parameters=dimension), or by "defining
equations" (i.e., as the solution space of a homogeneous linear system.)- examples of passing from one description to the other.
HW(5.5) 6c, d; 7(all); 8c, d; 9c,d ; 11b

9/26           QUIZ 3 (HW from 5.2, 5.3, 5.4)
solution of quiz 3
rank of  a matrix- definition, example (basis and defining equations for all 3 subspacees)
HW (5.6): 2d, 3d, 5, 7, 8

9/28               rank, bases, defining equations-more examples
orthogonal subspaces; orthogonal complement
the four subspaces associated to a matrix; orthogonality of row space and nullspace
tranpose matrix- connection with dot product
HW: (6.2): 13a,b,c, 14a, 15a, 16b,c

9/30          discussion of example from HW (6.2 14-15)
application of linear systems: linear optimization (intro)
Notes on linear optimization
(PDF file, 5 pages)
George B. Dantzig (1914-2005)

10/3             QUIZ 4 (HW from 5.6, 6.2)
linear optimization: the simplex method (solution of 1st problem in handout)
Note: the solution has been added to the handout as a 5th page.
HW: try the 2nd problem in the handout

10/5            Linear transformations defined geometrically
Examples in R^2 (rotations, projections, expansion/contraction): matrix in a given basis and
in the standard basis)
HW: 4.3: 12, 13, 17, 23

10/7              Coordinate changes- change of basis formulas for vectors and linear transformations
Example: projection onto a two-dimensional plane, parallel to a given direction
HW (8.4)4a, 6a, 9; (4.3):14
Note: the HW quiz next week will be on Wednesday, 10/12

10/10        Examples from HW for 4.3 and 8.4
Direct sum of two subspaces; general projections

10/12           QUIZ 5
Orthogonal projections (start)-projection onto a 1-dim subspace
HW: (6.3) 21, 22
Note:  although the topics presented in lecture are all in the book, I am not following the same order.
For this reason, I have been writing course notes that track the lectures more closely. For this part of the course, the
notes are found in the link above: "linear transformations defined geometrically". At the moment, they cover
the lectures from 10/5 to 10/10.

10/14              FALL BREAK- no class

10/17            Orthogonal projections (continued)

10/19             Review/questions

10/21            EXAM 2.
Sections included: 5.2, 5,3, 5,4, 5,5, 5,6: subspaces, bases, dimension, rank/nullity, row space-column space- nullspace
6.2: orthogonality, orthogonal complements, orthogonality of row space and nullspace
4.3 and 8.4:  matrices of  linear transformations defined geometrically
STUDY GUIDE: (1) problems from text assigned as homework (in the sections listed above); course notes
( "linear optimization" will not be on the test)
Exam 2(with solutions)-PDF file

10/24              orthogonal and orthonormal basis; Gram-Schmidt procedure, change of basis matrix in an o.n. basis.
HW  (6.3): 10, 13, 14, 16,  18,  22, 23

10/26:              least squares approximate solutions; normal system assoc. to a non-hom linear system
HW(6.4):  1a, 2a, 3b, 4b, 6, 9

10/28              QUIZ 6 (HW from 6.3)
least squares fitting to data
HW (9.3):  2, 4, 8
solutions to Exam 2  posted (see link above)

10/31               Orthogonal matrices; rotations in R^3
Products and inverses of orthogonal are orthogonal; examples in R^2: reflections, rotations
Example in R^3:  find the matrix for rotation with a given axis, by a given angle
HW (6.5): 16, 18

11/2                   Determinants: definition, main  properties; positively oriented bases of R^n (sect. 2.3)
Example: reflection across a given 2-dimensional subspace in R^3 (NOT IN TEXT)
Example: rotation in R^3 about a given axis, by a given angle (NOT IN TEXT)
Remark: the two topics above, not found in the text, are highly likely to appear in the next test, or the
ORTHOGONAL MATRICES

11/4                     QUIZ 7 (HW from 6.4, 9.3, 6.5)
eigenvalues/eigenvectors; diagonalizable matrices (start)
HW (7.1) 2a,b 3a,b 5a,b 6a,b 11, 12

11/7                       review of complex numbers (inc. the derivation of the DeMoivre-Laplace formula via power series)
HW (10.3): 14, 15, 20, 22

11/9                       COMPLEX EIGENVALUES OF REAL MATRICES
NOT IN TEXT! read this handout (PDF file) and do the problems proposed (= HW for this material)

The next HW quiz will be on Monday 11/14. (HW from 7.1, 7.2 and the 11/9 handout  = link above)
Exam 3 will be on Monday, 11/21

11/11                       diagonalizable matrices: examples for n=2 and n=3 (triangular). `Standard forms' of
diagonalizable matrices (n=2) with real or complex eigenvalues.
HW (7.2)3, 4, 5, 8, 9, 11

11/14                    QUIZ 8
Algebraic vs. geometric multiplicity- examples in dimension 3
1)  one real and two complex conjugate eigenvalues: standard form
2) three real eigenvalues: examples in the three non-diagonalizable cases

11/16                   Application:  Powers of matrices and systems of one-step difference equations
Reduction of two-step difference equations to one-step systems: solution of the Fibonacci recursion.
(NOT IN TEXT!)

11/18                    Example: system of difference equations defined by a 2X2 matrix with complex eigenvalues
preserving lines (real eigenvalues) vs. rotating lines (complex eigenvalues)
Review : eigenvalues of rotations, reflections, projections; reduction to standard form
inverse problem: constructing  matrices given geometric info

11/21                      EXAM 3.  Material included:  lectures from 10/17  to 11/18 (including material not in the text.)
Study in paricular: HW problems and ALL THE ONLINE HANDOUTS SINCE 10/24
(note: ORTHOGONAL MATRICES added 11/16- see above)
Important topics:  1) computing the matrix for orthogonal projection onto given subspaces (6.3)
2) orthonormal basis of a given subspace; Gram-Schmidt method (6.3)
3) least-squares solutions-relation with projections (6.4, 9.3)
4) orthogonal matrices: rotations, projections (6.5, handout)
5) eigenvalues; diagonalizable matrices (7.1, 7.2)
6) `standard form' of 2X2 matrices with complex eigenvalues (handout)
7) showing that a given matrix is NOT diagonalizable: examples when n=2,3  (7.2, lecture 11/14)
8) computing powers of a diagonalizable matrix; use to solve systems of difference equations.
(A test with six questions taken from these eight topics would be a safe bet.)

11/23                    Symmetric matrices- orthogonal diagonalization.
HW( 7.3)  1(c)(f), 2,5,9,11,12,14

11/25                    Thanksgiving (no classes)

11/28                    Quadratic forms: positive definiteness, extreme values, reduction to diagonal form, level sets
HW(9.5): 5b, 6a, 10, 11  HW(9.6): 1b, 4a, 6, 8, 11, 15

11/30                    The classification of matrices (n=2,3) (PDF file)
(includes 10 problems)

(This concludes the material for this courses. Friday and Monday will be review classes for the final, based on problems)
FINAL: Thursday, 12/15, 8-10 AM
Material: lectures of 11/23, 12/28 and 11/30, and last part of the handout "orthogonal matrices" (eigenvalues of orthogonal matrices and
of projections).
Eigenvalues, Google web searches, and college football
(Link to a short 2001 paper by Herbert Wilf (Penn)- inspirational reading, esp. for computer scientists!)

12/2                        Review problems: sections 7.3, 9.5, 9.6

12/5                        Review problems: from the 11/30 online handout and the list below:
Review problems (eigenvalues, powers) (with answers)

12/15                      Thursday, 12/15, 8:00-10:00 AM- FINAL EXAM (in the usual classroom)

TOPICS included in the final: symmetric matrices, quadratic forms, rotating conics into standard form
(7.3, 9.5, 9.6)
classification of matrices (standard forms in n=2,3)- online handout
eigenvalus of rotations, projections, reflections; powers of  matrices ( online handout `orthogonal matrices',
review problem set of 12/5 (link above)

Review: HW problems for the three sections listed, class notes, both online handouts listed (links above, 11/2 and 11/30),
review problems for 12/2 and 12/5.
Remark: a final page with answers to the problems has been added to the online handout of  11/30.

EXAM 4 ("final")