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Lin Alg & Matrices
Final Examination
Spring 1996
Your Name:
SOLUTION
For purpose of anonymous grading, please do
not
write your name on the subsequent pages.
This examination consists of 5 questions, each question counting for the given number of
points, adding to a total of 24 points. Please write your answers in the spaces indicated, or
below the questions (using the back of the sheets if necessary). You are allowed to consult
two
8
.
5
0
×
11
0
sheets with notes, but
not
your book or your class notes. If you get stuck on
a problem, it may be advisable to go to another problem and come back to that one later.
You will have
2 hours
to do this test.
Good luck!
Problem 1
2
3
4
5
Total
1
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View Full Document Problem 1
(7 points): Please answer the following questions and
justify your answers
brieﬂy.
(a, 2pts) Consider the following 2 problems: A. Given a basis for a subspace of
R
n
ﬁnd a basis for
its orthogonal complement. B. Given a matrix and one of its eigenvalues, ﬁnd a basis for
the corresponding eigenspace. Both problems can be reduced to the problem of ﬁnding
a basis for the nullspace of a matrix. Please explain what those nullspace problems look
like.
A. Let
v
1
,...,v
r
be a basis for
V
⊆
R
n
. Write
A
=[
v
1

v
2

...

v
r
]. Then
V
=
CS
(
A
),
hence
V
⊥
=
(
A
)
⊥
=
NS
(
A
T
).
B. The eigenspace is
(
A

λ
1
I
) where
λ
1
is the eigenvalue.
(b, 1pts) For the following weighted inner product on
R
3
,
h
x
,
y
i
=
x
1
y
1
+2
x
2
y
2
+
x
3
y
3
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This note was uploaded on 06/11/2010 for the course MA 405 taught by Professor Staff during the Spring '08 term at N.C. State.
 Spring '08
 STAFF

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