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CAAM 336
·
DIFFERENTIAL EQUATIONS
Examination 1
Posted Saturday, 16 October 2010.
Due no later than 5pm on Wednesday, 20 October 2010.
Instructions:
1. Time limit:
4 uninterrupted hours
.
2. There are four questions worth a total of 100 points, plus a 5point bonus.
Please do not look at the questions until you begin the exam.
3. You
may not
use any outside resources, such as books, notes, problem sets, friends,
calculators, or MATLAB.
4. Please answer the questions thoroughly and justify all your answers.
Show all your work to maximize partial credit.
5. Print your name on the line below:
6. Time started:
Time completed:
7. Indicate that this is your own individual eﬀort in compliance with the instructions above and
the honor system by writing out in full and signing the traditional pledge on the lines below.
8. Staple this page to the front of your exam.
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View Full DocumentCAAM 336
·
DIFFERENTIAL EQUATIONS
1. [25 points: 5 points per part]
For all problems below, assume
V
is a vector space endowed with an inner product, (
·
,
·
).
(a) Suppose
W
= span
{
φ
1
}
is a onedimensional subspace of
V
.
Write down a formula for the best approximation to
v
∈
V
from the subspace
W
.
In many circumstances, we are given a linearly independent basis
φ
1
,...,φ
N
for the subspace
W
= span
{
φ
1
,...,φ
N
}
,
and would like to create an orthogonal basis
ψ
1
,...,ψ
N
for this same subspace
W
.
This procedure is called the
Gram–Schmidt
process:
Set
ψ
1
=
φ
1
.
For
k
= 2
,...,N
Let
b
φ
k
denote the best approximation to
φ
k
from span
{
ψ
1
,...,ψ
k

1
}
.
Set
ψ
k
=
φ
k

b
φ
k
.
end
(b) Use the Gram–Schmidt process to construct orthogonal basis vectors
ψ
1
,ψ
2
,ψ
3
for the
subspace
W
= span
{
φ
1
,φ
2
,φ
3
}
of
R
4
, where
φ
1
=
1
0
0
0
,
φ
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 Fall '09
 Tompson

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