Assignment for Applied Linear Algebra  Math 270
February 11, 2011
Problem 1
(3 pts.)
Find the coefficients
b
1
, b
2
, b
3
∈
R
such that
sin(
t
) cos(2
t
) =
b
1
sin(
t
) +
b
2
sin(2
t
) +
b
3
sin(3
t
)
.
SOLUTION:
sin(
t
) cos(2
t
) =
e
it

e

it
2
i
e
i
2
t
+
e

i
2
t
2
=
1
4
i
(
e
i
3
t
+
e

it

e
it

e

i
3
t
)
=

1
2
e
it

e

it
2
i
+
1
2
e
i
3
t

e

i
3
t
2
i
.
Therefore:
b
1
=

1
2
,
b
2
= 0,
b
3
=
1
2
.
Problem 2
(9 pts.)
Let
P
1
be the vector space of real polynomials of degree less or equal than 1. Define the (nonlinear)
function
E
:
P
1
→
R
as
E
(
p
) =
Z
1
0
2
π
cos
πx
2

p
(
x
)
2
dx,
(where
p
=
p
(
x
) is a polynomial in
P
1
). Find the point of minimum for
E
, i.e., find the polynomial
q
∈
P
1
such that
E
(
q
)
≤
E
(
p
)
for all
p
∈
P
1
.
SOLUTION: For brevity, let
f
(
x
) :=
2
π
cos(
πx
2
), then, using the length
k · k
defined in the hint,
min
p
∈
P
1
E
(
p
) = min
p
∈
P
1
k
f

p
k
2
is given by the orthogonal projection of
f
on
P
1
. The projection can be computed in the following
way:
•
Find a basis of
P
1
, for example
{
1
, x
}
•
Reduce it to an equivalent orthonormal system
{
w
1
, w
2
}
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 Winter '09
 Linear Algebra, Algebra, Cos, Fast Fourier transform, πx

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