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MAT067
University of California, Davis
Winter 2007
Homework Set 7: More Exercises on Eigenvalues
Directions
: Please work on all of the following exercises. Submit your solutions to Problems
3(d) and 5(b) as your Calculational Problems and Problems 1 and 2 as your ProofWriting
Problems at the
beginning
of lecture on February 23, 2007.
As usual, we are using
F
to denote either
R
or
C
.
1. Let
a, b, c, d
∈
F
and consider the system of equations given by
ax
1
+
bx
2
=0
cx
1
+
dx
2
.
Note that
x
1
=
x
2
= 0 is a solution for any choice of
a, b, c
,and
d
. Prove that this
system of equations has a nontrivial solution if and only if
ad
−
bc
=0.
2. Let
A
=
±
ab
cd
²
∈
F
2
×
2
, and recall that we can de±ne a linear operator
T
∈L
(
F
2
)on
F
2
by setting
T
(
v
)=
Av
for each
v
=
±
v
1
v
2
²
∈
F
2
.
Show that the eigenvalues for
T
are exactly the
λ
∈
F
for which
p
(
λ
)=0
,wh
e
r
e
p
(
z
)=(
a
−
z
)(
d
−
z
)
−
bc
.
Hint
: Write the eigenvalue equation
Av
=
λv
as (
A
−
λI
)
v
= 0 and use Problem 1.
3. Find eigenvalues and associated eigenvectors for the linear operators on
F
2
de±ned by
the following 2
×
2 matrices:
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 Fall '07
 Schilling
 Linear Algebra, Algebra

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