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School
Rumson Fair Haven Reg H *
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Course
C101
Subject
Mathematics
Date
Nov 24, 2024
Type
Pages
9
Uploaded by CoachRiverTiger30
MA
265,
Fall
2022,
Midterm
II
(GREEN)
INSTRUCTIONS:
1.
Write
your
answers
of
the
seven
multiple
choice
questions
into
the
table
on
the
last
page.
Show
all
your
work
on
the
questions
and
you
may
use
the
back
of
the
test
pages
as
scratch
paper
if
needed.
2.
After
you
have
finished
the
exam, hand
in
your
test
booklet
to
your
instructor.
101
MWF
10:30AM
102
MWF
9:30AM
153
MWF
11:30AM
154
MWF
11:30AM
205
TR
1:30PM
206
TR
3:00PM
357
MWF
1:30PM
410
TR
1:30PM
451
TR
10:30AM
501
TR
12:00PM
502
MWF
10:30AM
Ying
Zhang
Ying
Zhang
Ying
Zhang
Danicl
Tuan-Dan
Le
Oleksandr
Tsymbaliuk
Oleksandr
Tsymbaliuk
Yiran
Wang
Arun
Debray
Arun
Debray
Vaibhav
Pandey
Ayan
Maiti
600
601
650
651
701
702
703
704
705
706
707
708
MWF
MWF
MWF
MWF
MWF
MWF
MWF
MWF
MWF
TR
MWF
MWF
1:30PM
1:30PM
10:30AM
9:30AM
3:30PM
11:30AM
12:30PM
1:30PM
12:30PM
1:30PM
3:30PM
2:30PM
Seongjun
Choi
Ayan
Maiti
Yevgeniya
Tarasova
Yevgeniya
Tarasova
Seongjun
Choi
Yiran
Wang
Ke
Wu
Ke
Wu
Seongjun
Choi
Vaibhav
Pandey
Siamak
Yassemi
Siamak
Yassemi
3.
NO
CALCULATORS,
BOOKS,
NOTES,
PHONES
OR
CAMERAS
ARE
ALLOWED
on
this
exam.
Turn
off
or
put
away
all
electronic
devices.
4.
When
time
is
called,
all
students
must
put
down
their
writing
instruments
immediately.
You
may
remain
in
your
seat
while
your
instructor
will
collect
the
exam
booklets.
5.
Anyone
who
violates
these
instructions
will
have
committed
an
act
of
academic
dis-
honesty.
Penalties
for
such
behavior
can
be
severe
and
may
include
an
automatic
F
on
the
course.
All
cases
of
academic
dishonesty
will
be
reported
to
the
Office
of
the
Dean
of
Students.
I
have
read
and
understand
the
above
instructions
regarding
academic
dishonesty:
STUDENT
NAME
STUDENT
SIGNATURE
STUDENT
PUID
SECTION
NUMBER
1.
(10
points)
Let
Lel
@
be
the
rank
of
A
and
b
be
the
nullity
of
A,
find
56
— 3a.
A
25
17
w0
1
0
e
{u,v,w}
is
a
basis
for
R®
provided
that
¢
is
not
equal
2
3
1
(10
points)
Let
u
=
{():|
V=
|:1}
,and
w
=
|:—1:[
where
¢
is
a
rcal
number.
The
sef
A
-2
B.
2
C.
-3
D.
3
E.
-1
3.
(10
points)
Which
of
the
following
statements
is
always
TRUE?
A.
If
Ax
=
Ax
for
some
veetor
X
,
then
A
is
an
cigenvalue
of
A.
If
v
is
an
cigenvector
corresponding
to
eigenvalue
2,
then
—v
is
an
cigenvector
corresponding
to
eigenvalue
—2.
C.
If
B
is
invertible.
then
matrices
A
and
B~'AB
could
have
different
sets
of
eigenval-
ues.
D.
If
Ais
an
cigenvalue
of
matrix
A,
then
A?
is
an
cigenvalue
of
matrix
A%
E.
If
-5
is
an
eigenvalue
of
matrix
B,
then
mateix
B
—
51
is
not
invertible.
4.
(10
poinis)
Let
P;
be
the
vector
space
of
all
polynomials
of
degrec
at
most
3.
Which
of
the
following
subsets
are
subspaces
of
Ps?
(i)
A
set
of
polynomials
in
Py
satisfying
p(0) =
p(1).
(il)
A
sct
of
polynomials
in
[Py
satisfying
p(0)p(1)
=
0.
(iif)
A
set
of
polynomials
in
P3
with
integer
cocfficients.
A.
(i)
only
B.
(i)
and
(ii)
only
C.
(i)
and
(iii)
only
(ii)
ouly
E.
(ii)
and
(jii)
only
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5.
(10
points)
Consider
the
differential
equation
[Bo)
=%
2
G
Then
the
origin
is
an
attractor
a
repeller
a
saddle
point
a
spiral
point
#
9
0w
>
none
of
the
above
6.
Which
of
the
following
matrices
are
diagonalizable
over
the
real
numbers?
(i)
and
(iii)
only
(iii)
and
(iv)
only
A
B
C.
(i),
(iii)
and
(iv)
only
D.
(i),
(ii)
and
(iii)
only
E
(i),
(i)
and
(iv)
only
7.
(10
points)
A
real
2
x
2
matrix
A
has
an
eigenvalue
A,
=
2
+
4
with
corresponding
eigenvector
v,
=
3~
4+l
Which
of
the
following
is
the
general
REAL
solution
to
the
system
of
differential
cquations
x'(t)
=
Ax(¢)?
B.
cpe?
et
o
2t
cre
e
[3cosi
—
sint]
e
[3sint
+
cost]
|4cost
+
sin
]
2
|4sint
—
cost]
[~3cost
+
sint
T
et
3sint
—
cost
|
4cosl
—sint
2
|
45int
—
cost
[3cost
—
sint]
[3sint
—
cost
o2t
|dcost
+
sint
]
+oe
|4sint
—
cost]
[3cost
+sint]
eyt
[3sint
+
cost)
|4cost
—sin|
2
|4sint
—
cost
[3cost
+ sint]
teget
[3sint
—
cost]
2
|[4cost
—sint)
|4sint
+
cost]
8.
Let
T':
Mayy
—
Mays
be
a
linear
map
defined
as
A
=
A
+
AT.
(2
points)
(1)
Find
’[’(
B
i]
)
{4
points)
(2)
Find
a
basis
for
the
range
of
T.
(4
points)
(3)
Find
a
basis
for
the
kernet
of
T.
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00
9.
(6
points)
(1)
Find
all
the
eigenvalues
of
matrix
A =
2
1},
and
find
a
basis
for
23
the
cigenspace
corresponding
to
cach
of
the
cigenvalues.
(4
points)
(2)
Find
an
invertible
matrix
P
and
a
diagonal
matrix
D such
that
4
00
1
2
1|
=pPDP
N
-1
2
3
10.
(4
points)
(1)
Find
the
eigenvalues
and
corresponding
eigenvectors
of
the
matrix
a=[8
4
(2
points)
(2)
Find
a
general
solution
to
the
system
of
differential
equations
o=
2]
5o
y(t)
ol
-
[
Gl
Bol-F)
(4
points)
(3)
Let
[T(t)]
be
a
particular
solution
to
the
initial
value
problem
Find
z(1)
+
y(1).
Please
write
your
answers
of
the
7
multiple
choice
questions
in
the
following
table.
Question
Answer
1.
(10
points)
2.
(10
points)
3.
(10
points)
4.
(10
points)
.
(10
points)
[4:4
6.
(10
points)
7.
(10
points)
Total
Points:
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