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MAS 3105
Nov 21, 2008
Quiz 6
Prof. S. Hudson
1) Let
V
=
C
[0
,
2
π
] with inner product
h
f,g
i
=
1
π
R
2
π
0
f
(
x
)
g
(
x
)
dx
.
a) Show that 1 is orthogonal to cos(5
x
) in
V
.
b) Compute

sin(3
x
)

. [If you have forgotten some Calculus/Trig skills, you can
replace sin(3
x
) by
x
2
, for partial credit].
2) Find the distance from the point P(1,2,3) to the plane
x
+
y
+ 2
z
= 0.
3) Choose ONE of these.
a) [based on MHW 4.1] Suppose that
F
=
{
w
1
,
w
2
,
w
3
}
=
{
e
3
,
e
1
+
e
2
,
e
1
}
is a
basis for
R
3
, and
L
:
R
3
→
R
3
. Suppose
L
(
w
1
) = 4
w
1
and
L
(
w
2
) = 4
w
1
+ 3
w
2
and
L
(
w
3
) = 4
w
1
+ 3
w
2
+ 2
w
3
. Find the matrix representation of
L
with respect to
F
.
b) Derive the formula for the projection matrix
P
for a Least Squares problem (it represents
projection onto
R
(
A
)). Then, show that
P
2
=
P
.
c) Show that if rank (
A
) =
n
(= number of columns), then the normal equations have a
unique solution ˆ
x
. (You may use the textbook proof, or repeat HW 5.2.13, etc)
Remarks and Answers:
The average was about 40/60. The scale for this quiz is: A’s
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 Spring '09
 JULIANEDWARDS

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