Math 414-501 Spring 2023
Test 1 Review
General Information
Test 1 will be given on Wednesday, 2/22/2023. Please bring an
8½×11 bluebook. I will have office hours on
Friday and Monday, 10-12 and 3-4, and on Tuesday, 10-11 and 3-5.
-
Calculators. 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 or storing programs or
other material.
-
Other devices. You may not use cell phones, computers, or any
other device capable of storing, sending, or receiving information.
-
Structure and coverage. There will be 4 to 6 questions, some
with multiple parts. The test will cover sections 0.1-0.5 and
1.2.1-1.2.4 in the text. The problems will be similar to ones done for
homework, and as examples in class and in the text. A
short table
of integrals and Fourier transform properties will be provided. Here
are links to practice
tests:
2002, 2009 and 2015
Topics Covered
Inner Product Spaces
- Inner products
- Definitions of real and complex inner products, examples of inner
product spaces.
- Standard inner products
on Rn, Cn, L2 and
ℓ 2, various examples. Be able to determine whether or
not YTAX is a real inner product, given A. §0.2, §0.3
- Be able to compute the angle between two vectors, the length of a
vector, and the distance between two vectors. §0.4.
- Types of convergence (See Fourier series below.) Be able
to define each type and to explain the differences between them.
- Pointwise convergence. §0.3.1
- Mean convergence. §0.3.1.
- Uniform convergence. §0.3.2.
- Orthogonality
- Orthogonal and orthonormal sets of vectors, orthonormal bases,
and orthogonal complements of subspaces. Know the definitions for
these terms. Know how to write a vector in terms of an orthonormal
basis, and how to calculate the coefficients. Be able to do problems
similar to ones assigned in homework. §0.5.1
- Orthogonal projections and least squares. Be able to find
orthogonal projections and to solve minimization
problems and least-squares fitting problems. §0.5.2
- Gram-Schmidt process. Be able to find an o.n. set from a given
non-orthogonal set. §5.3
Fourier Series
- Calculating Fourier Series
- Extensions of functions periodic, even periodic, and odd
periodic extensions. Be able to sketch extensions of functions.
- Fourier series. Be able to compute Fourier series in either real
or complex forms, and with prescribed period 2$\pi$ on an intervals of
the form $[-\pi, \pi]$, $[0, 2\pi]$. Be able to know and use Lemma 1.3.
- Fourier sine series (FS for odd, 2$\pi$-periodic extension) and
Fourier cosine series (FS for even, 2$\pi$-periodic extension). Be
able to compute FSS and FCS for functions defined on a half interval,
$[0, \pi]$.
- Be able to use symmetry properties to help compute coefficients
in FS, FSS, FSC.
- Partial sums
- Be able to show that the partial sum $S_N(x)$ of the FS for a
function $f$ is the projection of $f$ onto the span of $\{1,\cos(x),
\sin(x), \cos(2x),\sin(2x),\ldots, \cos(Nx), \sin(Nx)\}$. Notes 2/10/23.
- Pointwise Convergence of Fourier series
- Know the conditions under which the partial sum of an
FS, FSS, FCS is pointwise convergent. Be able to use them to decide
what function an FS, FSS, or FCS converges to pointwise. §1.2.4