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Rumson Fair Haven Reg H *
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Course
C101
Subject
Mathematics
Date
Nov 24, 2024
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14
Uploaded by CoachRiverTiger30
MA26500
FINAL EXAM
Fall 2022
GREEN - Test Version 01
NAME
INSTRUCTOR
1. NO CALCULATORS, BOOKS, NOTES, PHONES OR CAMERAS ARE ALLOWED
on this exam. Use the back of the test pages for scratch paper.
PLEASE PUT YOUR
SCANTRON UNDERNEATH YOUR QUESTION SHEETS WHEN YOU
ARE NOT FILLING IN THE SCANTRON SHEET.
2. Fill in your name and your instructor’s name on the question sheets (above).
3. You must use a
#2 pencil
on the mark–sense sheet (answer sheet). Fill in the instruc-
tor’s name and the course number(
MA265
), fill in the correct TEST/QUIZ NUMBER
(
GREEN
is 01), your name, your section number (see below if you are not sure), and
your 10-digit PUID(BE SURE TO INCLUDE THE TWO LEADING ZEROS of your
PUID.) and shade them in the appropriate spaces. Sign the mark–sense sheet.
101
MWF
10:30AM
Ying Zhang
600
MWF
1:30PM
Seongjun Choi
102
MWF
9:30AM
Ying Zhang
601
MWF
1:30PM
Ayan Maiti
153
MWF
11:30AM
Ying Zhang
650
MWF
10:30AM
Yevgeniya Tarasova
154
MWF
11:30AM
Daniel Tuan-Dan Le
651
MWF
9:30AM
Yevgeniya Tarasova
205
TR
1:30PM
Oleksandr Tsymbaliuk
701
MWF
3:30PM
Seongjun Choi
206
TR
3:00PM
Oleksandr Tsymbaliuk
702
MWF
11:30AM
Yiran Wang
357
MWF
1:30PM
Yiran Wang
703
MWF
12:30PM
Ke Wu
410
TR
1:30PM
Arun Debray
704
MWF
1:30PM
Ke Wu
451
TR
10:30AM
Arun Debray
705
MWF
12:30PM
Seongjun Choi
501
TR
12:00PM
Vaibhav Pandey
706
TR
1:30PM
Vaibhav Pandey
502
MWF
10:30AM
Ayan Maiti
707
MWF
3:30PM
Siamak Yassemi
708
MWF
2:30PM
Siamak Yassemi
4. There are 25 questions, each is worth 8 points. Show your work on the question sheets.
also
CIRCLE
your answer choice for each problem on the question sheets in case your
scantron is lost. Although no partial credit will be given, any disputes about grades or
grading will be settled by examining your written work on the question sheets.
5.
Please remain seated during the last 10 minutes of the exam.
When time is
called, all students must put down their writing instruments immediately.
6.
Turn in both the mark-sense sheets and the question sheets to your instruc-
tor when you are finished
.
1
1.
Let
A
be a 4
×
4 matrix. Which of the following statements is always TRUE?
A.
The reduced echelon form of
A
has at least 1 pivot.
B.
If
A
2
= 0 then the system
A
x
= 0 has only the trivial solution.
C.
If
A
2
=
A
then the reduced echelon form of
A
is
I
4
.
D.
If
A
2
=
I
4
then the reduced echelon form of
A
is
I
4
.
E.
A
is diagonalizable over the complex numbers.
2.
For which of the following five values of the parameter
a
is the set
(
a
a
a
,
1
a
a
,
2
3
a
)
linearly independent?
(i)
a
= 0
(ii)
a
= 1
(iii)
a
= 2
(iv)
a
= 3
(v)
a
= 4
A.
(iv) and (v) only
B.
(iii) and (v) only
C.
(i), (ii), and (iv) only
D.
(ii), (iii), and (iv) only
E.
(i), (ii), (iii), and (iv) only
2
3.
Let
T
:
R
n
→
R
m
be a linear transformation and
T
(
x
) =
A
x
, where
A
is the standard
matrix for
T
. Which of the following statements must be TRUE?
(i)
A
is an
n
×
m
matrix.
(ii) If the columns of
A
are linearly independent, then
T
is one-to-one.
(iii) If
m > n
, then
T
is onto.
(iv) If
n > m
, then
T
is one-to-one.
(v) If the columns of
A
span
R
m
, then
T
is onto.
A.
(i), (ii), (iii), (iv), and (v)
B.
(i), (ii), and (iii) only
C.
(ii) and (v) only
D.
(ii), (iii), and (iv) only
E.
(iii) and (iv) only
4.
Consider the system of linear equations
x
+ 3
y
-
z
= 5
2
x
+ 5
y
+
az
= 9
x
+
y
+
a
2
z
=
a
Under which conditions does this system have infinitely many solutions?
A.
a
6
=
-
1
B.
a
6
= 3
C.
a
=
-
1
D.
a
= 3
E.
a
6
=
-
1 and
a
6
= 3
3
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5.
Which of the following statements is always TRUE?
A.
Let
x
,
y
,
z
∈
R
3
. If
z
is not a multiple of
y
, and
y
is not a multiple of
x
, then
{
x
,
y
,
z
}
is linearly independent.
B.
Let
T
:
R
n
→
R
m
be a one-to-one linear transformation.
If
{
x
,
y
}
is a linearly
independent set in
R
n
, then so is
{
T
(
x
)
, T
(
y
)
}
in
R
m
.
C.
If
A
is an
m
×
n
matrix and
m
≥
n
, then
A
has linearly independent columns.
D.
If
A
is an
m
×
n
matrix and
A
x
=
b
is consistent for every
b
∈
R
m
, then the columns
of
A
are linearly independent.
E.
If the columns of a matrix
A
are linearly independent, then the equation
A
x
=
b
has
a unique solution for every
b
.
6.
Let
L
:
R
2
→
R
3
be a linear transformation such that
L
-
1
2
!
=
1
-
1
-
2
,
L
2
-
3
!
=
-
1
2
3
,
and
L
a
b
!
=
1
3
2
.
Then
a
+
b
=
A.
5
B.
-
5
C.
1
D.
-
1
E.
9
4
7.
Let
A
=
1
1
1
1
1
-
1
0
3
4
2
2
2
2
2
3
.
Which of the following statements is FALSE?
A.
3
-
4
1
0
0
,
4
-
5
0
1
0
is a basis for Nul(
A
).
B.
A
x
=
2
-
1
4
is consistent.
C.
1
-
1
2
,
1
0
2
,
1
2
3
is a basis for Col(
A
).
D.
1
-
1
2
,
1
3
2
,
1
2
3
is a basis for Col(
A
).
E.
1
-
1
2
,
1
0
2
,
1
4
2
is a basis for Col(
A
).
8.
Let
A
=
1
0
1
1
0
2
-
1
3
1
and
B
=
A
-
1
. Find
b
12
, the (1
,
2)-entry of
B
.
A.
0
B.
2
C.
-
2
D.
-
1
E.
1
5
9.
Which of the following sets of vectors
x
y
z
is a subspace of
R
3
?
(i) The set of all vectors satisfying
x
+
y
+
z
= 0.
(ii) The set of all vectors satisfying
xy
-
z
= 0.
(iii) The set of all vectors satisfying
xyz
= 0.
(iv) The set of all vectors satisfying
x
+
y
-
z
2
= 0
.
(v) The set of all vectors satisfying
x
+
y
-
z
≥
0.
A.
(i) only
B.
(i), (iii), (iv), and (v) only
C.
(i) and (ii) only
D.
(i) and (v) only
E.
None of them is a subspace of
R
3
10.
The points (1
,
2), (2
,
4), (4
,
5), and (5
,
7) are the vertices of a parallelogram on the coor-
dinate plane. What is the area of this parallelogram?
A.
-
3
B.
3
C.
6
D.
0
E.
1
6
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11.
If
a
b
c
d
e
f
g
h
k
=
-
5
,
find
2
d
-
3
f
e
f
2
a
-
3
c
b
c
2(
a
+
g
)
-
3(
c
+
k
)
b
+
h
c
+
k
.
A.
5
B.
-
5
C.
10
D.
-
10
E.
30
12.
Let
A
=
1
-
1
0
2
2
0
1
5
-
1
1
0
3
. Which of the following statements is always TRUE?
A.
Col
(
A
) is a subspace of
R
4
.
B.
Nul
(
A
) is a subspace of
R
3
.
C.
The equation
A
x
= 0 has a unique solution.
D.
The dimension of the null space of
A
is 1.
E.
The rank of the matrix
A
is 2.
7
13.
If the characteristic polynomial of
A
is 9
λ
-
λ
3
, then
A.
A
is invertible and diagonalizable.
B.
A
is diagonalizable, but not invertible.
C.
A
is invertible, but not diagonalizable.
D.
A
is neither invertible nor diagonalizable.
E.
A
is invertible, but may or may not be diagonalizable.
14.
The mapping
T
:
P
2
→
P
2
defined by
T
(
a
0
+
a
1
t
+
a
2
t
2
) =
a
1
+ 2
a
2
t
is a linear transfor-
mation. Find [
T
]
B
, the
B
-matrix for
T
when
B
is the basis
{
1
, t, t
2
}
for
P
2
.
A.
[
T
]
B
=
0
2
0
0
0
1
0
0
0
B.
[
T
]
B
=
0
0
0
0
1
0
0
0
2
C.
[
T
]
B
=
0
0
0
0
2
0
0
0
1
D.
[
T
]
B
=
1
1
0
1
2
2
0
0
0
E.
[
T
]
B
=
0
1
0
0
0
2
0
0
0
8
15.
Let the transformation
T
:
x
→
A
x
be the composition of a rotation by angle
θ
∈
(
-
π, π
],
followed by a scaling by the factor
r >
0. If
A
=
-
9
√
3
9
-
9
-
9
√
3
, find
θ
and
r
.
A.
θ
=
-
5
π/
6
,
r
= 18
B.
θ
=
π/
3
,
r
= 9
C.
θ
= 5
π/
6
,
r
= 18
D.
θ
=
π/
6
,
r
= 18
E.
θ
=
-
π/
6
,
r
= 18
16.
Consider the following system of differential equations
x
0
(
t
) = 4
x
(
t
) + 2
y
(
t
)
y
0
(
t
) = 2
x
(
t
) + 4
y
(
t
)
with the initial condition
x
(0) = 8
, y
(0) = 2. What is the value of
x
(1) +
y
(1)?
A.
10
e
6
B.
10
e
-
6
C.
6
e
6
D.
10
e
6
+ 6
e
2
E.
6
e
6
+ 10
e
2
9
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17.
If
v
is an eigenvector of
A
with eigenvalue 2 and also an eigenvector of
B
with eigenvalue
3, then
v
is an eigenvector of
A
2
-
BA
+ 3
I
with eigenvalue
A.
5
B.
-
5
C.
1
D.
-
1
E.
-
2
18.
Consider the real solution to the initial value problem
(
x
0
(
t
) =
A
x
(
t
)
,
x
(0) =
x
0
,
where
x
(
t
) =
x
1
(
t
)
x
2
(
t
)
,
A
=
0
2
-
8
0
, and
x
0
=
1
2
. Then
x
1
(
π
4
) is
A.
2
B.
0
C.
1
D.
-
1
E.
-
2
10
19.
Which of the following statements is FALSE?
A.
If rows of an
m
×
n
matrix
A
are orthonormal, then the linear transformation
x
→
A
x
preserves lengths.
B.
If columns of a
k
×
m
matrix
A
are orthonormal and columns of an
m
×
n
matrix
B
are orthonormal, then the columns of their product
AB
are also orthonormal.
C.
If
A
is an orthogonal
n
×
n
matrix and
B
is an
n
×
n
matrix obtained from
A
by
interchanging some rows, then
B
is also orthogonal.
D.
If columns of an
m
×
n
matrix
A
are orthonormal, then the linear transformation
x
→
A
x
preserves lengths.
E.
If
A
and
B
are two orthogonal
n
×
n
matrices, then
AB
T
is also orthogonal.
20.
Let
u
1
=
1
2
2
,
u
2
=
5
1
1
,
y
=
9
2
-
2
. Then the closest point ˆ
y
in
W
= Span
{
u
1
,
u
2
}
to
y
is:
A.
ˆ
y
=
4
-
1
-
1
B.
ˆ
y
=
0
0
0
C.
ˆ
y
=
9
0
0
D.
ˆ
y
=
28
/
3
11
/
3
11
/
3
E.
ˆ
y
=
9
-
2
2
11
21.
Which of the following statements is FALSE?
A.
If the set
{
u
,
v
,
w
}
is orthonormal, then the set
{
u
+
v
,
u
-
v
,
w
}
is orthogonal.
B.
If a matrix
A
has orthogonal columns, then
A
T
A
is a diagonal matrix.
C.
Orthogonal matrices are invertible.
D.
For vectors
u
and
v
, if
k
u
-
v
k
2
=
k
u
k
2
+
k
v
k
2
, then
u
and
v
are orthogonal.
E.
Performing the Gram-Schmidt process on the set
{
v
1
,
v
2
}
and the set
{
v
2
,
v
1
}
always
gives the identical orthonormal basis for the Span
{
v
1
,
v
2
}
.
22.
The least-squares solution of an inconsistent system
1
0
1
0
0
1
x
=
1
0
0
is
A.
ˆ
x
=
1
0
B.
ˆ
x
=
1
2
1
0
C.
ˆ
x
=
1
1
D.
ˆ
x
=
1
3
1
1
E.
ˆ
x
=
1
2
1
1
12
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23.
If one orthogonally diagonalizes
A
=
3
1
1
3
=
PDP
T
, then
P
and
D
can be chosen as
A.
P
=
1
/
√
2
-
1
/
√
2
1
/
√
2
1
/
√
2
D
=
2
0
0
4
B.
P
=
1
/
√
2
1
/
√
2
-
1
/
√
2
1
/
√
2
D
=
2
0
0
4
C.
P
=
1
/
√
2
-
1
/
√
2
1
/
√
2
1
/
√
2
D
=
-
2
0
0
-
4
D.
P
=
5
/
√
26
7
/
√
50
1
/
√
26
1
/
√
50
D
=
-
2
0
0
-
4
E.
P
=
-
5
/
√
26
7
/
√
50
1
/
√
26
-
1
/
√
50
D
=
2
0
0
4
24.
Let
C
[
-
3
,
3] be the vector space of all continuous real-valued functions defined on [
-
3
,
3]
.
Define the inner product on
C
[
-
3
,
3] by
h
f, g
i
=
Z
3
-
3
f
(
t
)
g
(
t
)
dt.
Let
W
be the subspace spanned by the polynomials 1 and
t
. Find the orthogonal projec-
tion of
t
2
+
t
onto
W
with respect to the above inner product on
C
[
-
3
,
3]
.
A.
3 +
t
B.
3
-
t
C.
t
2
+
t
D.
3 + 2
t
E.
3
-
2
t
13
25.
Consider the basis
S
for
R
3
given by
1
1
0
,
1
0
1
,
0
0
1
.
If we apply the Gram–Schmidt process to
S
to obtain an orthonormal basis, we obtain
A.
1
√
2
1
1
0
,
1
√
2
1
-
1
0
,
0
0
1
B.
1
√
2
1
1
0
,
1
3
√
2
1
-
1
4
,
1
3
2
-
2
-
1
C.
1
√
2
1
1
0
,
1
3
√
2
-
1
1
4
,
1
3
2
-
2
1
D.
1
√
2
1
1
0
,
1
√
3
1
-
1
1
,
1
√
6
-
1
1
2
E.
1
√
2
1
1
0
,
1
√
6
1
-
1
2
,
1
√
3
-
1
1
1
14