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
Exam
1-problems
Exam
1-solutions
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
reading assignment: p.117-121
problems(ch. 6): 24,27 (try it for the shift in l^2)
28,29,32
6/26
adjoints; self-adjoint and normal operators
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)
