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MAT067
University of California, Davis
Winter 2007
Solutions to Homework Set 7
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
(1)
cx
1
+
dx
2
.
(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.
Solution
: Assume that
ad
−
bc
6
= 0 and consider the combination
d
(1)
−
b
(2),
which can be written as (
ad
−
bc
)
x
1
= 0. Since by assumption the factor (
ad
−
bc
)
doesn’t vanish, we must conclude that
x
1
.
Now consider the combination
−
c
(1) +
a
(2). This similarly yields (
ad
−
bc
)
x
2
=0,
so, again using the assumption, we must conclude that
x
2
as well.
Therefore, assuming
ad
−
bc
6
,wehavefoundtha
t
x
1
=
x
2
= 0 is the unique
solution.
If on the other hand,
ad
−
bc
,then
x
1
=
d, x
2
=
−
c
and
x
1
=
−
b, x
2
=
a
are
solutions. If at least one of
a, b, c, d
is nonzero, this yields a nontrivial solution. If
a
=
b
=
c
=
d
= 0, then certainly all
x
1
,x
2
∈
F
satisfy (1) and (2).
2. Let
A
=
±
ab
cd
²
∈
F
2
×
2
, and recall that we can deFne 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.
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Solution
: One method for Fnding the eigeninformation of
T
is to analyze the
solutions of the matrix equation
Av
=
λv
for
λ
∈
F
and
v
∈
F
2
. In particular,
using the deFnition of eigenvector and eigenvalue,
v
is an eigenvector associated to
the eigenvalue
λ
if and only if
Av
=
T
(
v
)=
λv
.
A simpler method involves the equivalent matrix equation (
A
−
λI
)
v
=0
,where
I
denotes the identity map on
F
2
. In particular, 0
6
=
v
∈
F
2
is an eigenvector for
T
associated to the eigenvalue
λ
∈
F
if and only if the system of linear equations
(
a
−
λ
)
v
1
+
bv
2
cv
1
+(
d
−
λ
)
v
2
±
(3)
has a nontrivial solution. Moreover, the system of equations (3) has a nontrivial
solution if and only if the polynomial
p
(
λ
)=(
a
−
λ
)(
d
−
λ
)
−
bc
evaluates to zero
(see Problem 1).
In other words, the eigenvalues for
T
are exactly the
λ
∈
F
for which
p
(
λ
)=0
,
and the eigenvectors for
T
associated to an eigenvalue
λ
are exactly the nonzero
vectors
v
=
²
v
1
v
2
³
∈
F
2
that satisfy the system of equations (3).
3. ±ind eigenvalues and associated eigenvectors for the linear operators on
F
2
deFned by
each given 2
×
2 matrix.
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This homework help was uploaded on 04/08/2008 for the course MATH 67 taught by Professor Schilling during the Fall '07 term at UC Davis.
 Fall '07
 Schilling
 Linear Algebra, Algebra, Equations

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