Math 311-503 - Final Exam Review
General Information
The Final Exam will be held in our usual classroom on Friday,
May 6, from 12:30 to 2:30. Please bring two
8½×11 bluebook. You may use calculators to do
arithmetic, although you will not need them. You may not use
any calculator that has the capability of doing linear algebra.
The test will have 7 to 9 questions, some with multiple parts. The
test will directly cover material from the following sections of the
book: 3.3-3.7, 14.6-8. It will also cover material from my notes
listed below:
Material from chapters 1 and 2 and sections 3.1-3.2 will not be
covered directly, although you will need to know it to answer
questions on the material listed above. Problems will be similar to
ones done for homework or as examples in class or in the
notes. Here are links to a practice exam and extra problems.
Material that you haven't been tested over will be weighted
more heavily than old. Thus, expect sections 3.3-3.6A to be about 35%,
and the rest to be about 65%, give or take a few percent. For a review
of those sections, look at problems from Test II. The other sections
are dealt with below.
Eigenvalues and Eigenvectors
- Eigenvalues and eigenvectors. Be able to find the
eigenvalues and eigenvectors for a linear transformation or matrix,
and be able to solve simple ODE problems.
- Diagonalization. In the case of an n×n
matrix A, we say that A is diagonalizable if and only there is a basis
for Rn (or Cn) composed of
eigenvectors of A. When this happens, we can change to the basis of
eigenvectors using an invertible matrix S = [x1
... xn], where the columns of S are the linearly
independent eigenvectors of A. We get a diagonal matrix D =
S-1AS. The diagonal entries in D are the eigenvalues of A
listed in the same order as the eigenvectors are in the columns of
S. Be able to find S if A is diagonalizable. Also, for a general linear
transformation L:V->V, we say that L is diagonalizable if and only if
there is a basis for V relative to which the matrix for L is
diagonal. It is possible to diagonalize L be finding a matrix for it
relative to some basis and then diagonalizing that matrix.
Inner products and norms
- Inner product, angle, and norm. For a given inner
product < , >, be able to find the norm of a
vector and to find the angle between two vectors. Know the four
properties that define an inner product - positivity, symmetry,
homogeneity, and additivity. Be able to use them in calculations.
- Orthogonality. Know and understand these terms:
orthogonal and orthonormal sets; orthonormal bases. Be able to use the
Gram-Schmidt procedure to find an orthogonal or orthonormal set of
vectors, given both an inner product and a linearly independent
set. Be able to do simple least-squares problems, either discrete or
continuous. Be able to explain the connection between least-squares
property and partial sums of Fourier series.
Fourier Series and Special Functions
- Fourier series. Be able to compute Fourier series
for simple functions, and be able to sketch the function to which the
Fourier series converges pointwise.
- Bessel functions and Legendre polynomials. Be able
to solve Bessel's equation or similar equations using the method of
Frobenius. Know how to show a differential operator is self
adjoint. Be able to show that eigenfunctions of a self adjoint
operator corresponding to distinct eigenvalues are orthogonal. Be able
to use this to show that Legendre polynomials are orthogonal relative
to the inner product
Also, be able to do problems similar to ones on the practice exam and
on the sheet with addition problems.
Updated: 5/3/05