FINAL EXAM PRACTICE I, May 8, 2010
Math 21b, Spring 10
Name:
MWF10 Ryan Reich
MWF10 Stewart Wilcox
MWF11 Oliver knill
MWF11 Ryan Reich
MWF11 Stewart Wilcox
MWF12 Juliana Belding
TTH10 Wushi Goldring
TTH1130 Alex Subotic
•
Please write your name above and check your sec
tion to the left.
•
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
next empty page for work. If you need additional
paper, write your name on it.
•
Do not detach pages from this exam packet or un
staple the packet.
•
Please write neatly and except for problems 13,
give details and justifications. Answers which are
illegible for the grader can not be given credit.
•
No notes, books, calculators, computers, or other
electronic aids can be allowed.
•
You have 180 minutes time to complete your work.
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20
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10
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10
4
10
5
10
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10
7
10
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10
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13
10
Total:
140
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Problem 1) (20 points) True or False? No justifications are needed.
1)
T
F
If
A
is a real
n
×
n
matrix, then
A
+
A
T
is diagonalizable.
2)
T
F
The equation
z
3
=
−
1 has three different complex solutions
z
and
z
=
−
1
is one of them.
3)
T
F
There is a 2
×
2 projection matrix
A
that projects onto a line and a 2
×
2
reflection matrix
B
that rotates about a point, so that
AB
is a rotation
matrix.
4)
T
F
The transformation
T
(
f
)(
x
) =
f
(
f
(
x
)) is linear on the space
C
∞
of all
smooth functions
5)
T
F
If a continuous function
f
defined on the interval [
−
π, π
] is both even and
odd, then it must be constant function.
6)
T
F
If
A
is a 2
×
2 matrix, then the characteristic polynomial of
AA
T
is
λ
2
+
cλ
for some real constant
c
.
7)
T
F
The function
f
(
t
) =
e
t
is an eigenfunction with eigenvalue 1 of the linear
operator
T
=
D
2
where
Df
=
f
′
.
8)
T
F
The initial value problem
f
′′
(
x
) +
f
(
x
) = sin(
x
)
, f
′′
(0) = 1
, f
′
(0) = 1 has
exactly one solution.
9)
T
F
The transformation
L
(
A
) =
A
+
A
T
is a linear transformation on
M
n
and
L
has only the eigenvalues 0 and 2.
10)
T
F
The set
X
of smooth functions
f
(
x, y, z
) which satisfy the
f
xx
+
f
yy
+
f
zz
=
f
is a linear space.
11)
T
F
If
A
is a 5
×
5 matrix that has rank 2, then the eigenvalue 0 has algebraic
multiplicity 3.
12)
T
F
Every equilibrium point of a nonlinear system ˙
x
=
f
(
x, y
)
,
˙
y
=
g
(
x, y
) is
located on at least one nullcline.
13)
T
F
The transformation
T
(
A
) = rank(
A
) is linear from the space
M
2
of all real
2
×
2 matrices to
R
.
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 Spring '10
 smith
 Math, Linear Algebra, Algebra, Matrices, Eigenvalue, eigenvector and eigenspace

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