Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
1)
A mathematician has found 4 solutions to a homogeneous system of 36 equations in 39
variables. The 4 solutions are linearly independent and all other solutions can be
constructed by adding together appropriate multiples of these 4 solutions. Will the system
necessarily have a solution for every possible choice of constants on the right side of the
equation? Explain.
1)
2)
Suppose a nonhomogeneous system of 13 linear equations in 16 unknowns has a solution
for all possible constants on the right side of the equation. Is it possible to find 5 nonzero
solutions of the associated homogeneous system that are linearly independent? Explain.
2)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Determine whether {v1, v2, v3} is a basis for
3.
3)
v1=1
–3
4,v2=–3
8
–6, v3=2
–2
–4
A)
No
B)
Yes
Find a basis for the column space of the matrix.
4)
Find a basis for Col B where
B =
1–1 0 –2 0 0
0 0 1 4 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 0 0 0 0
.
A)
1
0
0
0
0
,
0
0
1
0
0
,
0
0
0
1
0
B)
1
0
0
0
,
0
1
0
0
,
0
0
1
0
,
0
0
0
1
C)
1
0
0
0
0
,
–1
0
0
0
0
,
0
1
0
0
0
,
–2
4
0
0
0
,
0
0
1
0
0
,
0
0
0
1
0
D)
1
0
0
0
0
,
0
1
0
0
0
,
0
0
1
0
0
,
0
0
0
1
0
Solve the problem.
5)
If the null space of a 7×5 matrix is 2–dimensional, find Rank A, Dim Row A, and Dim Col A.
A)
Rank A =3, Dim Row A =3, Dim Col A =2
B)
Rank A =3, Dim Row A =3, Dim Col A =3
C)
Rank A =5, Dim Row A =5, Dim Col A =5
D)
Rank A =3, Dim Row A =2, Dim Col A =2
Find an explicit description of the null space of matrix A by listing vectors that span the null space.
6)
A =1–2–5–3
0 1 1 2
A)
2
1
0
0
,
5
–1
1
0
,
3
–2
0
1
B)
2
1
0
0
,
3
–1
1
0
,
–1
–2
0
1
C)
3
–1
1
0
,
–1
–2
0
1
D)
5
–1
1
0
,
3
–2
0
1
Solve the problem.
7)
Find all values of h such that y will be in the subspace of
3 spanned by v1, v2, v3 if v1=1
2
–4,
v2=3
4
–8,v3=–1
0
0, and y=7
7
h.
A)
h = –28
B)
h = –14 or 0
C)
all h –14
D)
h = –14
Use coordinate vectors to determine whether the given polynomials are linearly dependent in P2. Let B be the standard
basis of the space P2 of polynomials, that is, let B=1, t, t2.
8)
1 + 2t, 3 + 6t2, 1 + 3t + 4t2
A)
Linearly dependent
B)
Linearly independent
Determine which of the sets of vectors is linearly independent.
9)
A: The set sin t , tan t in C[0, 1]
B: The set sin t cos t , cos 2t in C[0, 1]
C: The set cos2 t , 1 + cos 2t in C[0, 1]
A)
A and B
B)
C only
C)
B only
D)
A and C
E)
A only
For the given matrix A, find k such that Nul A is a subspace of
k and find m such that Col A is a subspace of
m.
10)
A = 4 0 0 –1 1 –7
2 6 –5–1 0 3
–3–4 4 –3 2 5
10)
A)
k = 3, m = 3
B)
k = 3, m = 6
C)
k = 6, m = 6
D)
k = 6, m = 3
Solve the problem.
11)
Let H =
a +3b +4d
c + d
–3a –9b + 4c –8d
–c – d
: a, b, c, d in
Find the dimension of the subspace H.
11)
A)
dim H = 4
B)
dim H = 3
C)
dim H = 2
D)
dim H = 1
Use coordinate vectors to determine whether the given polynomials are linearly dependent in P2. Let B be the standard
basis of the space P2 of polynomials, that is, let B=1, t, t2.
12)
1 + 2t +t2, 3 –6t2, 1 + 4t +4t2
12)
A)
Linearly dependent
B)
Linearly independent
Find the specified change–of–coordinates matrix.
13)
Consider two bases B=b1, b2 and C=c1, c2 for a vector space V such that
b1=c1–2c2 and b2=3c1–4c2. Find the change–of–coordinates matrix from B to C.
13)
A)
1–2
3–4
B)
0 3
–2–4
C)
1 3
2 4
D)
1 3
–2–4
Find a basis for the set of all solutions to the difference equation.
14)
yk+3–2yk+2–9yk+1+18yk= 0 for all k
14)
A)
3k, (–2)k, (–3)k
B)
2k, 3k, (–3)k
C)
2k, 3k
D)
2k, 3k, (–3)k, (–4)k
4
Solve the problem.
15)
Determine which of the following sets is a vector space.
V is the line y = x in the xy–plane: V =x
y : y = x
W is the union of the first and second quadrants in the xy–plane: W =x
y : y 0
U is the line y = x + 1 in the xy–plane: U =x
y : y = x + 1
15)
A)
U only
B)
W only
C)
V only
D)
U and V
Find a matrix A such that W = Col A.
16)
W =–2s –3t
2t
3s + t : s, t in
16)
A)
–2 2 3
3 0 1
B)
–2 0 3
–3 2 1
C)
–2–3
0 2
D)
–2 3
2 0
Find the steady–state probability vector for the stochastic matrix P.
17)
P =0.2 0.5
0.8 0.5
17)
A)
5
8
B)
1
5
4
5
C)
8
13
5
13
D)
5
13
8
13
Determine whether the signals are linearly independent.
18)
2k – 5, 7k –1, k +2
18)
A)
Yes
B)
No
5
Determine whether the set of vectors is a basis for
3.
19)
Given the set of vectors 1
0
0, 0
1
2, decide which of the following statements is true:
A: Set is linearly independent and spans
3. Set is a basis for
3.
B: Set is linearly independent but does not span
3. Set is not a basis for
3.
C: Set spans
3 but is not linearly independent. Set is not a basis for
3.
D: Set is not linearly independent and does not span
3. Set is not a basis for
3.
19)
A)
C
B)
A
C)
B
D)
D
Solve the problem.
20)
Suppose that the weather in a certain city is either sunny, cloudy, or rainy on a given day, and
consider the following:
1) If it is sunny today there is a 70% chance it will be sunny tomorrow and a 30% chance that it will
be cloudy.
2) If it is cloudy today there is a 40% chance it will be sunny tomorrow, a 40% chance that it will be
cloudy, and a 20% chance that it will be rainy.
3) If it is rainy today there is a 40% chance it will be sunny tomorrow, a 30% chance that it will be
cloudy, and a 30% chance that it will be rainy.
In the long run, how likely is it that the weather will be rainy on a given day?
20)
A)
11.3%
B)
12.1%
C)
10.6%
D)
9.5%
Find the dimensions of the null space and the column space of the given matrix.
21)
A =
1–2 3 1 0 5 –4
0 0 1 –6 2 –2 0
0 0 0 0 0 1 3
0 0 0 0 0 0 0
21)
A)
dim Nul A = 2, dim Col A = 5
B)
dim Nul A = 4, dim Col A = 3
C)
dim Nul A = 3, dim Col A = 4
D)
dim Nul A = 5, dim Col A = 2
Determine whether the vector u belongs to the null space of the matrix A.
22)
u=–1
–1
1, A =–2–3–5
–4–2–6
3 –2 3
22)
A)
Yes
B)
No
Solve the problem.
23)
Let v1=–2
–2
2,v2=–3
1
–2, v3=6
–10
14 , and H = Span v1, v2, v3 .
Note that v3=3v1–4v2. Which of the following sets form a basis for the subspace H, i.e., which
sets form an efficient spanning set containing no unnecessary vectors?
A: v1, v2, v3
B: v1, v2
C: v1, v3
D: v2, v3
23)
A)
A only
B)
B only
C)
B, C, and D
D)
B and C
Find the vector x determined by the given coordinate vector [x]B and the given basis B.
24)
B= 1
–3
4,–3
8
4, 2
–2
–1 , [x]B=–2
–3
2
24)
A)
11
–22
–22
B)
7
–22
1
C)
0
–9
14
D)
7
–11
–22
Find the coordinate vector [x]B of the vector x relative to the given basis B.
25)
b1=3
5
–4, b2=4
–3
–2, x=6
–19
2, and B =b1, b2
25)
A)
42
–38
–12
B)
–2
3
C)
–58
87
14
D)
–2
2
If the set W is a vector space, find a set S of vectors that spans it. Otherwise, state that W is not a vector space.
26)
W is the set of all vectors of the form
a –5b
2
6a + b
–a – b
, where a and b are arbitrary real numbers.
26)
A)
1
2
6
–1
,
–5
0
1
–1
B)
1
0
6
–1
,
–5
0
1
–1
,
0
2
0
0
C)
1
0
6
–1
,
–5
2
1
–1
D)
Not a vector space
Find an explicit description of the null space of matrix A by listing vectors that span the null space.
27)
A =1–2 3 –3–1
–2 5 –5 4 2
–1 3 –2 1 1
27)
A)
–5
–1
1
0
0
,
7
–2
0
1
0
,
0
0
0
0
1
B)
2
1
0
0
0
,
–3
–1
1
0
0
,
3
2
0
1
0
,
–1
0
0
0
1
C)
–5
–1
1
0
0
,
7
2
0
1
0
,
1
0
0
0
1
D)
1
0
5
–7
–1
,
0
1
1
–2
0
Find a matrix A such that W = Col A.
28)
W =
3r – t
4r – s +3t
s +3t
r –5s + t
: r, s, t in
28)
A)
0 3–1
4–1 3
0 1 3
1 –5 1
B)
3 4 0 1
0 –1 1 –5
–1 3 3 1
C)
3 0 –1
4–1 3
0 1 3
1 –5 1
D)
3–1
4 3
1 3
1 –5
Find the specified change–of–coordinates matrix.
29)
Let B=b1, b2 and C=c1, c2 be bases for
2, where
b1=1
–3, b2=–3
3, c1=1
3, c2=–4
–10 .
Find the change–of–coordinates matrix from B to C.
29)
A)
1–4
3 –10
B)
1–3
–3 6
C)
–11 21
–3 6
D)
– 2 7
– 1 11
3
Find the coordinate vector [x]B of the vector x relative to the given basis B.
30)
b1=1
2, b2=1
–4, x=1
8, and B =b1, b2
30)
A)
2
0
B)
2
–1
C)
1
8
D)
9
–30
Find the new coordinate vector for the vector x after performing the specified change of basis.
31)
Consider two bases B=b1, b2, b3 and C=c1, c2, c3 for a vector space V such that
b1=c1+2c3, b2=c1+4c2–c3, and b3=3c1–c2.Suppose x=b1+6b2+b3. That is,
suppose [x]B=1
6
1.Find [x]C.
31)
A)
10
25
–6
B)
10
23
–4
C)
10
23
8
D)
3
24
–3
Solve the problem.
32)
Let H be the set of all polynomials of the form p(t) = a + bt2 where a and b are in
and b > a.
Determine whether H is a vector space. If it is not a vector space, determine which of the following
properties it fails to satisfy.
A: Contains zero vector
B: Closed under vector addition
C: Closed under multiplication by scalars
32)
A)
H is not a vector space; does not contain zero vector
B)
H is not a vector space; not closed under vector addition
C)
H is not a vector space; not closed under multiplication by scalars and does not contain zero
vector
D)
H is not a vector space; not closed under multiplication by scalars
Determine whether the set of vectors is a basis for
3.
33)
Given the set of vectors 1
0
0, 0
1
0, 0
0
1, 0
1
1, decide which of the following statements is true:
A: Set is linearly independent and spans
3. Set is a basis for
3.
B: Set is linearly independent but does not span
3. Set is not a basis for
3.
C: Set spans
3 but is not linearly independent. Set is not a basis for
3.
D: Set is not linearly independent and does not span
3. Set is not a basis for
3.
33)
A)
A
B)
C
C)
D
D)
B
Solve the problem.
34)
If A is a 5×9 matrix, what is the smallest possible dimension of Nul A?
34)
A)
9
B)
0
C)
4
D)
5
Determine if the vector u is in the column space of matrix A and whether it is in the null space of A.
35)
u=5
–3
5 , A = 1 –3 4
–1 0 –5
3 –3 6
35)
A)
Not in Col A, in Nul A
B)
In Col A, not in Nul A
C)
Not in Col A, not in Nul A
D)
In Col A and in Nul A
Find the new coordinate vector for the vector x after performing the specified change of basis.
36)
Consider two bases B=b1, b2 and C=c1, c2 for a vector space V such that
b1=c1–6c2 and b2=4c1+3c2. Suppose x=b1+2b2. That is, suppose [x]B=1
2. Find [x]C.
36)
A)
9
0
B)
–11
10
C)
6
–9
D)
8
0
Determine whether the signals are linearly independent.
37)
1k, 2k, (–3)k
37)
A)
No
B)
Yes
For the given matrix A, find k such that Nul A is a subspace of
k and find m such that Col A is a subspace of
m.
38)
A =
1–2
0 6
–4 5
–1–3
4 1
38)
A)
k = 2, m = 5
B)
k = 5, m = 5
C)
k = 5, m = 2
D)
k = 2, m = 2
Find a basis for the column space of the matrix.
39)
Let A =
–1 3 7 2 0
1 –2–7–1 3
2 –4–9–5 1
3 –6–11 –9–1
and B =
1–3–7–2 0
0 1 0 1 3
0 0 5 –3–5
0 0 0 0 0
.
It can be shown that matrix A is row equivalent to matrix B. Find a basis for Col A.
39)
A)
–1
1
2
3
,
3
–2
–4
–6
,
2
–1
–5
–9
B)
1
0
0
0
,
–3
1
0
0
,
–7
0
5
0
C)
–1
1
2
3
,
3
–2
–4
–6
,
7
–7
–9
–11
D)
–1
1
2
3
,
3
–2
–4
–6
,
7
–7
–9
–11
,
2
–1
–5
–9
,
0
3
1
–1
Solve the problem.
40)
Determine which of the following statements is false.
A: The dimension of the vector space P7 of polynomials is 8.
B: Any line in
3 is a one–dimensional subspace of
3.
C: If a vector space V has a basis B =b1, ..…., b3, then any set in V containing 4 vectors must be
linearly dependent.
40)
A)
A
B)
C
C)
A and B
D)
B
41)
Suppose that demographic studies show that each year about 6% of a city’s population moves to
the suburbs (and 94% stays in the city), while 4% of the suburban population moves to the city (and
96% remains in the suburbs). In the year 2000, 60.7% of the population of the region lived in the city
and 39.3% lived in the suburbs. What is the distribution of the population in 2002? For simplicity,
ignore other influences on the population such as births, deaths, and migration into and out of the
city/suburban region.
41)
A)
55.7% in the city and 44.3% in the suburbs
B)
58.6% in the city and 41.4% in the suburbs
C)
57.7% in the city and 42.3% in the suburbs
D)
56.8% in the city and 43.2% in the suburbs
Find the vector x determined by the given coordinate vector [x]B and the given basis B.
42)
B=1
3, 0
1 , [x]B=–4
4
42)
A)
1
4
B)
4
–8
C)
–4
4
D)
–4
–8
Determine whether {v1, v2, v3} is a basis for
3.
43)
v1=–2
4
4, v2=1
0
–5, v3= 4
–4
–14
43)
A)
Yes
B)
No
Solve the problem.
44)
Determine which of the following sets is a subspace of Pn for an appropriate value of n.
A: All polynomials of the form p(t) = a + bt2, where a and b are in
B: All polynomials of degree exactly 4, with real coefficients
C: All polynomials of degree at most 4, with positive coefficients
44)
A)
B only
B)
A only
C)
A and B
D)
C only
Determine if the vector u is in the column space of matrix A and whether it is in the null space of A.
45)
u=
–4
3
–4
–2
, A =
1 0 3
–2–1–4
3 –3 0
–1 3 6
45)
A)
Not in Col A, in Nul A
B)
In Col A and in Nul A
C)
Not in Col A, not in Nul A
D)
In Col A, not in Nul A
If the set W is a vector space, find a set S of vectors that spans it. Otherwise, state that W is not a vector space.
46)
W is the set of all vectors of the form
a +6b
5b
5a – b
–a
, where a and b are arbitrary real numbers.
46)
A)
1
5
5
–1
,
6
0
–1
0
B)
1
0
5
–1
,
6
5
–1
0
C)
1
0
5
0
,
6
0
–1
0
,
0
5
0
–1
D)
Not a vector space
Assume that the matrix A is row equivalent to B. Find a basis for the row space of the matrix A.
47)
A =
1 3 –4 0 1
2 4 –5 5 –2
1 –5 0 –3 2
–3–1 8 3 –4
, B =
1 3 –4 0 1
0 –2 3 5–4
0 0 –8–23 17
0 0 0 0 0
47)
A)
{(1, 3, –4, 0, 1), (2, 4, –5, 5), –2, (1, –5, 0, –3, 2), (–3, –1, 8, 3, –4)}
B)
{(1, 3, –4, 0, 1), (0, –2, 3, 5, –4), (0, 0, –8, –23, 17), (0, 0, 0, 0, 0)}
C)
{(1, 0, 0, 0), (3, –2, 0, 0), (–4, 3, –8, 0)}
D)
{(1, 3, –4, 0, 1), (0, –2, 3, 5, –4), (0, 0, –8, –23, 17)}
Find the general solution of the difference equation.
48)
Given that the signal yk= 3k –2 is a solution of the given difference equation, find a description of
all solutions of the equation.
yk+2–6yk+1+5yk= –12 for all k
48)
A)
3k –2+c1+c2(5)k
B)
3k –2+c1+c2(6)k
C)
3k –2+c1(–1)k+c2(–5)k
D)
c1+c2(5)k
Determine which of the sets of vectors is linearly independent.
49)
A: The set p1, p2, p3 where p1(t) = 1, p2(t) =t2, p3(t) =2+3t
B: The set p1, p2, p3 where p1(t) = t, p2(t) =t2, p3(t) =2t +3t2
C: The set p1, p2, p3 where p1(t) = 1, p2(t) =t2, p3(t) =2+3t +t2
49)
A)
all of them
B)
A only
C)
C only
D)
A and C
E)
B only
Find a basis for the set of all solutions to the difference equation.
50)
yk+2+3yk+1–10yk= 0 for all k
50)
A)
(–2)k, 5k
B)
2k, (–6)k
C)
2k, (–5)k
D)
2k, (–2)k, (–5)k
Find the general solution of the difference equation.
51)
Given that the signal yk=k2 is a solution of the given difference equation, find a description of all
solutions of the equation.
yk+2+2yk+1–3yk=8k +6 for all k
51)
A)
c1+c2(–3)k
B)
k2+c1+c2(–3)k
C)
k2+c1(–1)k+c2(3)k
D)
k2+c1+c2(–2)k
Solve the problem.
52)
Let H be the set of all polynomials having degree at most 4 and rational coefficients. Determine
whether H is a vector space. If it is not a vector space, determine which of the following properties
it fails to satisfy.
A: Contains zero vector
B: Closed under vector addition
C: Closed under multiplication by scalars
52)
A)
H is not a vector space; not closed under vector addition
B)
H is not a vector space; not closed under multiplication by scalars
C)
H is a vector space.
D)
H is not a vector space; does not contain zero vector
Find the steady–state probability vector for the stochastic matrix P.
53)
P =0.9 0.3 0.2
0.1 0.6 0.7
0 0.1 0.1
53)
A)
30
41
10
41
1
41
B)
3
5
3
10
1
10
C)
29
39
3
13
1
39
D)
29
9
1
Solve the problem.
54)
Let H be the set of all points of the form (s, s–1). Determine whether H is a vector space. If it is not a
vector space, determine which of the following properties it fails to satisfy.
A: Contains zero vector
B: Closed under vector addition
C: Closed under multiplication by scalars
54)
A)
H is not a vector space; not closed under vector addition
B)
H is a vector space.
C)
H is not a vector space; does not contain zero vector
D)
H is not a vector space; fails to satisfy all three properties
55)
Suppose that the weather in a certain city is either sunny, cloudy, or rainy on a given day, and
consider the following:
1) If it is sunny today there is a 70% chance it will be sunny tomorrow and a 30% chance that it will
be cloudy.
2) If it is cloudy today there is a 40% chance it will be sunny tomorrow, a 40% chance that it will be
cloudy, and a 20% chance that it will be rainy.
3) If it is rainy today there is a 40% chance it will be sunny tomorrow, a 30% chance that it will be
cloudy, and a 30% chance that it will be rainy.
Suppose the predicted weather for Friday is 55% sunny, 35% cloudy, and 10% rainy. What are the
chances that Sunday will be rainy?
55)
A)
10%
B)
8.7%
C)
11%
D)
9.7%
56)
Determine which of the following statements is true.
A: If V is a 4–dimensional vector space, then any set of exactly 4 elements in V is automatically a
basis for V.
B: If there exists a set v1, ..…., v5 that spans V, then dim V =5.
C: If H is a subspace of a finite–dimensional vector space V, then dim H dim V.
56)
A)
A and C
B)
B
C)
C
D)
A
Determine whether the vector u belongs to the null space of the matrix A.
57)
u=2
3
1, A =–2 3 –5
–2–210
57)
A)
Yes
B)
No
Solve the problem.
58)
Suppose that demographic studies show that each year about 6% of a city‘s population moves to
the suburbs (and 94% stays in the city), while 2% of the suburban population moves to the city (and
98% remains in the suburbs). In the year 2000, 61.2% of the population of the region lived in the city
and 38.8% lived in the suburbs. What percentage of the population of the region would eventually
live in the city if the migration probabilities were to remain constant over many years? For
simplicity, ignore other influences on the population such as births, deaths, and migration into and
out of the region.
58)
A)
37.5%
B)
75.0%
C)
25.0%
D)
50%
59)
Let H be the set of all points in the xy–plane having at least one nonzero coordinate:
H =x
y : x, y not both zero . Determine whether H is a vector space. If it is not a vector space,
determine which of the following properties it fails to satisfy:
A: Contains zero vector
B: Closed under vector addition
C: Closed under multiplication by scalars
59)
A)
H is not a vector space; fails to satisfy all three properties
B)
H is not a vector space; not closed under vector addition
C)
H is not a vector space; does not contain zero vector and not closed under multiplication by
scalars
D)
H is not a vector space; does not contain zero vector
Find the dimensions of the null space and the column space of the given matrix.
60)
A = 1 –5–4 3 0
–2 3 –1–4 1
60)
A)
dim Nul A = 2, dim Col A = 3
B)
dim Nul A = 3, dim Col A = 2
C)
dim Nul A = 3, dim Col A = 3
D)
dim Nul A = 4, dim Col A = 1
Answer Key
Testname: C4
20
Answer Key
Testname: C4