MATH662: Quiz 1
Answer the following questions in no more than 10 minutes.
1. Do n linearly independent vectors v1,
., vn in a vector space (V , R, +) form a basis?
2. Is the set of 2 2 diagonal matrices a subspace of the vector space of 2 2 matrices?
3.
MATH662 Midterm Examination 03/06/14
Answer the following questions explaining all steps that lead to a solution.
1. Let A Cmm. Use the singular value decomposition of A to nd the eigenvalue decomposition
of
M=
0 A
A 0
.
Compare: (a) use of Rayleigh quoti
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<section|MATH662 Lecture 4 (1/21/14)>
<subsection|Projection>
<\remark>
One way to mathematically express the very valuable concept of dimension
reduction (reduced modeling, approximate modeling) is
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<section|MATH662 Homework 6, Issued 3/28/14, Due 4/3/14>
1. In eigenproblem algorithms we often reduce the size of a matrix after
finding one of its eigenvalue/vector pairs through a process known as
<em|deflatio
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<section|MATH662 Homework 7, Issued 4/3/14, Due 4/10/14>
1. Problem 36.1 from Trefethen & Bau.
<em|Solution>. At step <math|n> the incomplete factorization of
<math|A\<in\>\<bbb-R\><rsup|m\<times\>m>, <math|A> s.
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<section|MATH662 Homework 3, Issued 2/5/14, Due 2/13/14>
1. Complete the proof from notes in Lect08 to obtain the condition number
of computing root <math|j> under modifications of coefficient <math|i> of
polynom
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<section|MATH662 Lecture 3 (1/16/14)>
<subsection|Matrix norms>
<\remark>
The norm induced by the scalar product is widely used (Euclidean
distance), but other types of norms are very useful in anal
MATH662: Quiz 1
Answer the following questions in no more than 10 minutes.
., vn in a vector space (V , R, +) form a basis?
1. Do n linearly independent vectors v1,
Only if spancfw_v1, , vn = V
2. Is the set of 2 2 diagonal matrices a subspace of the vec
MATH662 Homework 1, Issued 1/16/14, Due 1/23/14
1. Let entries of A Cm n , B Cn p depend on x. Prove:
d
dA
dB
(A(x)B (x) =
B+A
dx
dx
dx
Let m = n, A be nonsingular. Prove
d
dA 1
(A 1(x) = A1(x)
A (x)
dx
dx
Solution . Using the repeated index summation con
MATH662: Quiz 2
Answer the following questions in no more than 10 minutes.
1. Can you interpret a vector u Cm as a function? If so, dene the three components of the function.
2. Are the functions cfw_1, x, x2,
, xn ,
orthogonal?
3. What is the scalar pro
CHAPTER 11
Finite element methods
1. Preliminaries
For a number of applications the restrictions imposed by nite dierence or
spectral methods with respect to the computational grid are too severe. This is
especially the case in structural engineering wher