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CAAM 336
·
DIFFERENTIAL EQUATIONS
Problem Set 4
Posted Wednesday 15 September 2010. Due Wednesday 22 September 2010, 5pm.
1. [20 points]
The equation
x
1
+
x
2
+
x
3
= 0 deﬁnes a plane in
R
3
that passes through the origin.
(a) Find two linearly independent vectors in
R
3
whose span is this plane.
(b) Find the point in this plane closest (in the standard Euclidean norm,
k
z
k
=
√
z
T
z
) to the vector
v
=
1
0
1
by formulating this as a best approximation problem. (You may use MATLAB to invert a matrix.)
2. [25 points]
Recall that a linear operator
P
is a projection from the vector space
V
to the vector space
V
provided
P
2
=
P
, that is,
P
(
Pf
) =
Pf
for all
f
∈
V
. Consider
V
=
C
[

1
,
1] with the usual inner product
(
u,v
) =
Z
1

1
u
(
x
)
v
(
x
)
dx,
and the two linear operators
P
e
and
P
o
the project a function onto their even and odd parts. That is,
(
P
e
f
)(
x
) =
f
(
x
) +
f
(

x
)
2
,
(
P
o
f
)(
x
) =
f
(
x
)

f
(

x
)
2
.
(a) Show that
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 Fall '09
 Tompson

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