Math 414-501 Test 1 Review
General Information
Test 1 will be given on Wednesday, 2/24/10. Please bring an
8½×11 bluebook. I will have office hours on
Monday (2/22), 2-3, and on Tuesday (2/23), 11:30-2.
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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.
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Other devices. You may not use cell phones, computers, or any
other device capable of storing, sending, or receiving information.
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Structure and coverage. There will be 4 to 6 questions, some
with multiple parts. The test will cover sections 0.1-0.5, 1.2, and
1.3.1-1.3.3 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 will be provided.
Topics Covered
Inner Product Spaces
- Inner products
- Definitions of real and complex inner products. Schwarz and
triangle inequalities. Be able to compute the angle between two
vectors. §0.2, §0.4.
- Examples of inner product spaces. ℝn,
ℂn, L2 and
ℓ2. §0.2, §0.3.
- Types of convergence
- Pointwise convergence. fn(x) → f(x)
for each fixed x. §0.3.1
- Convergence in L2 (mean convergence). ||f−
fn||L2 → 0. §0.3.1.
- Uniform convergence. maxx|fn(x) −
f(x)| → 0. §0.3.2.
- Know the definitions for these types of convergence. Be able to
give examples illustrating the differences among them.
- Orthogonality
- Orthogonal and orthonormal sets of vectors, orthonormal bases,
and orthogonal complements, V⊥. 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" minimization
problems. Be able to find orthogonal projections and to solve
least-squares minimization problems. §0.5.2.
- Gram-Schmidt process. Be able to use this to find a an
orthonormal basis for a space. §0.5.3.
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 2π on an intervals of
the form [−π, π], [0, 2π], or [c − π, c +
π]. Be able to know and use Lemma 1.3.
- Fourier sine series (FS for odd, 2π-periodic extension) and
Fourier cosine series (FS for even, 2π-periodic extension). Be
able to compute FSS and FCS for functions defined on a half interval,
[0,π].
- Pointwise convergence
- Riemann-Lebesgue Lemma. Be able to give a proof of this in the
simple case that f is continuously differentiable. §1.3.1.
- Fourier (Dirichlet) kernel, P. Know what P is and how to express
partial sums in terms of P. §1.3.2
- Be able to sketch a proof of Theorem 1.22, making use of the
formula for P and the properties of P as well as the Riemann-Lebesgue
Lemma. §1.3.2.
- Know the Theorems 1.22 and 1.28. Be able to use them to decide
what function an FS, FSS, or FCS converges to.
Updated 2/20/2010.