MATH 453- SUMMER 2002- COURSE LOG

5/30   vector space over a field, subspaces
product and quotient spaces
sums and direct sums of subspaces

problems: (ch. 1): 9,12,13,14,15

5/31     spanning sets, l.i. and l.d . sets, linear span
finite-dimensional spaces: existence of a basis
definition of dimension
criterion for infinite-dimensionality

problems (ch. 2): 6,9,11,13,14

6/3          subspace of f.d. is f.d.
existence of complements, dimension of V+W
from text: 12,15

6/4          Linear maps, linear operators, kernel and range
rank + nullity formula
from text: 2, 9,24

problems: 1,3,4,10,11,16 (ch.3)

6/5         linear operators and matrices
change of basis formula
similarity of matrices, conjugacy of operators

problems: 22,23,25,26 (ch. 3)

6/6           problems from chapter 3
invariant subspaces
eigenvectors, eigenspaces

problems: 4,5,7,8,11,12 (ch.5)

6/7           problems from ch. 3
eigenvalues: examples (inc. projections)
linear indep. of eigenspaces
existence of eigenvalues; triangularizable operators

problems: 14,15,16,20,23 (ch.5)

6/10       operators on complex v.sp. are triangularizable
eigenvalues of triangular matrices
diagonalizable operators
eigenspaces, similarity and conjugacy
problems from ch.5: 7,8,9,11,12,15,16

6/11 Exam 1
Topics: ch.1,2,3,5 (up to p.90 only)
Review: all problems listed above
summary1.ps
summary1.pdf

6/12         `generalized kernel' of an operator
Theorem: basis-independent description of `algebraic multiplicity'
nilpotent operators

problems: 3,4,5,8,11,12

6/13           problems from ch. 8
generalized eigenspaces, algebraic multiplicities
Theorem: existence of a basis of generalized eigenvectors
A=S+N,  S diagonalizable, N nilpotent, SN=NS

problems: 12 (the example)13, 15,16
reading assignment: characteristic polynomial, Cayley-Hamilton (p.172-173)
minimal polynomial (p.179-182)

6/14           Theorem: standard form for nilpotent operators
chains of generalized eigenvectors
Jordan standard form on complex vector spaces

reading assignment: square roots, p. 177-178
problems (ch. 8): 18,19,20, 22,26,27

6/17            Jordan form: examples
Complexification of real vector spaces and operators
Real Jordan form: diagonalizable case
problems: try some from ch.8 or ch. 9- none due tomorrow

6/18             real Jordan form: general case
exponentials of operators: computation using Jordan forms
application: fundamental solution for systems of ODEs
problems (ch.9): 8,9,12,13
(solve using the theory seen in class)

6/19              Review problems: Jordan form, exponentials, cyclic vectors/rational form

6/20  EXAM 2: Ch.8,Ch.9 (inc. material seen in class and reading assignments)
Exam 2-problems
solutions-page 1
solutions-page 2

6/21            Normed vector spaces [not in text!];
continuity and equivalence of norms in R^n
completeness, def. of Banach space, examples;
exponentials of operators
HOMEWORK PROBLEMS : 2 chosen from problems given in class

6/24              inner-product spaces, Hilbert spaces
orthogonal complements, projections, minimizing property

problems(ch.6): 5,6,7,8,17,18,21,22 (choose 3)

6/25               problems:  completeness of l^p ,
polarization identity,  norms satisfying the parallelogram law ,
characterizations of orthogonal projections, minimization in L^2

problems(ch. 6): 24,27 (try it for the shift in l^2) 28,29,32

spectral theorem for normal operators on hermitian vector spaces

problems (ch.7):   2,3,4,7,8,9

6/27                spectral theorem: self-adjoint operators and normal operators on real vector spaces
quadratic form associated to a self-adjoint operator
positive operators, isometries and the polar decomposition
problems from chapter 7

problems (ch 7): 11,13,14,17,22,23,24 (for review only)
solutions to some review problems

6/28 EXAM 3
Topics: ch.6, ch.7, normed vector spaces
The `summary of results' has been updated and may be used during the test.
EXAM 3-PROBLEMS
Exam 3-solutions

Course grades: A,  B+,  B+,  B,  B,  C,  C       (7 students took the last exam)