Math 54 Sample Final Exam
In this exam, the following formulas were given:
Z
e
ax
sin
bx dx
=
e
ax
(
a
sin
bx

b
cos
bx
)
a
2
+
b
2
+
C
Z
e
ax
cos
bx dx
=
e
ax
(
a
cos
bx
+
b
sin
bx
)
a
2
+
b
2
+
C
x
∼
∞
X
n
=1
2(

1)
n
+1
n
sin
nx ,

π < x < π
or 0
< x < π
x
∼
π
2

4
π
∞
X
n
=1
1
(2
n

1)
2
cos
nx ,
0
< x < π
1.
(12 points) Find the inverse of the matrix
A
=
7
10

9
1
2

3

1
1

6
, if it exists.
Use the
algorithm from the book (or from class).
2.
(20 points) Let
A
be the 2
×
2 matrix
0
1
x
0
, where
x
is a real number.
(a). For which values of
x
is
A
similar to a (real) diagonal matrix? (Do not diagonalize
the matrix.)
(b). For which values of
x
is
A
orthogonally diagonalizable?
3.
(20 points) Each of the following parts gives vector spaces
V
and
W
, bases
B
for
V
and
C
for
W
, and a linear transformation
T
:
V
→
W
. In each case find the matrix for
T
relative
to
B
and
C
.
(a).
V
=
W
=
R
2
,
B
=
{
(1
,
1)
,
(

1
,
1)
}
,
C
=
{
(1
,
0)
,
(0
,
1)
}
, and
T
is counterclockwise
rotation by 90 degrees.
(b).
V
=
W
= Span
{
sin
x,
cos
x
} ⊆
C
[0
,
2
π
],
B
=
C
=
{
sin
x,
cos
x
}
, and
T
is the linear
transformation taking a function to its derivative.
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 Spring '08
 Chorin
 Linear Algebra, Formulas, Diagonal matrix, 90 degrees, eax sin bx, 7y