Math3EHW#4
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Determine which of the following sets is a subspace of P n for an appropriate value of n.
A: All polynomials of the form p(t) = a + bt2, where a and b are in ℛ
B: All polynomials of degree exactly 4, with real coefficients
C: All polynomials of degree at most 4, with positive coefficients
A) B only
B) C only
C) A and B
D) A only
2) Determine which of the following sets is a vector space.
x : y=x
V is the line y = x in the xy-plane: V =
y
W is the union of the first and second quadrants in the xy-plane: W =
U is the line y = x + 1 in the xy-plane: U =
A) U only
B) U and V
x : y = x +1
y
C) V only
1)
2)
x : y≥0
y
D) W only
3) 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
A) H is not a vector space; not closed under vector addition
B) H is not a vector space; does not contain zero vector
C) H is a vector space.
D) H is not a vector space; not closed under multiplication by scalars
3)
4) Let H be the set of all polynomials of the form p(t) = a + bt2 where a and b are in ℛ 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
A) H is not a vector space; does not contain zero vector
B) H is not a vector space; not closed under multiplication by scalars
C) H is not a vector space; not closed under vector addition
D) H is not a vector space; not closed under multiplication by scalars and does not contain
zero vector
4)
1
5) 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
A) H is not a vector space; does not contain zero vector
B) H is not a vector space; fails to satisfy all three properties
C) H is a vector space.
D) H is not a vector space; not closed under vector addition
5)
6) Let H be the set of all points in the xy-plane having at least one nonzero coordinate:
x : x, y not both zero . Determine whether H is a vector space. If it is not a vector
H=
y
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
A) H is not a vector space; does not contain zero vector
B) H is not a vector space; fails to satisfy all three properties
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; not closed under vector addition
6)
Determine which of the sets of vectors is linearly independent.
7) 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
A)
B)
C)
D)
E)
7)
A and C
A only
C only
B only
all of them
8) 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 cos 2 t , 1 + cos 2t
in C[0, 1]
A) A only
B) A and C
8)
C) A and B
2
D) B only
E) C only
Determine whether the set of vectors is a basis for ℛ3.
1
0
0
0
9) Given the set of vectors
, decide which of the following statements is
0 , 1 , 0 , 1
0
0
1
1
true:
A: Set is linearly independent and spans ℛ3 . Set is a basis for ℛ3 .
9)
B: Set is linearly independent but does not span ℛ3 . Set is not a basis for ℛ3 .
C: Set spans ℛ3 but is not linearly independent. Set is not a basis for ℛ3 .
D: Set is not linearly independent and does not span ℛ3 . Set is not a basis for ℛ3 .
A) A
B) D
C) C
D) B
Solve the problem.
10) Let H =
a + 3b + 4d
c +d
: a, b, c, d in ℛ
-3a - 9b + 4c - 8d
-c - d
10)
Find the dimension of the subspace H.
A) dim H = 4
B) dim H = 2
C) dim H = 1
D) dim H = 3
Find an explicit description of the null space of matrix A by listing vectors that span the null space.
11) A = 1 -2 -5 -3
0 1 1 2
A)
B)
5
3
2
5
3
-1 , -2
1 , -1 , -2
1
0
0
1
0
0
1
0
0
1
C)
D)
2
3
-1
3
-1
1 , -1 , -2
-1 , -2
0
1
0
1
0
0
0
1
0
1
1 -2 3 -3
12) A = -2 5 -5 4
-1 3 -2 1
A)
-5
7
-1
2
1 , 0 ,
0
1
0
0
C)
2
-3
1
-1
0 , 1 ,
0
0
0
0
-1
2
1
11)
12)
B)
1
0
0
0
1
1
0
0
1
5 , 1
-7
-2
-1
0
D)
3
-1
2
0
0 , 0
1
0
0
1
-5
7
0
-1
-2
0
1 , 0 , 0
0
1
0
0
0
1
3
Find a basis for the column space of the matrix.
13) Find a basis for Col B where
1
0
B= 0
0
0
A)
-1
0
0
0
0
0 -2
1 4
0
0
0 0
0
0
0
0
1
0
0
13)
0
0
0 .
1
0
B)
1
0
0
0
0
0
0 , 1 , 0
0
0
1
0
0
0
1
0
0
0
0 , 1 , 0 , 0
0
0
1
0
0
0
0
1
C)
D)
1
0
0
0
0
1
0
0
,
,
,
0
0
1
0
0
0
0
1
0
0
0
0
-1
14) Let A = 1
2
3
3
-2
-4
-6
7
-7
-9
-11
1
0
0 ,
0
0
2
-1
-5
-9
0
3 and B =
1
-1
1
0
0
0
-3
1
0
0
-7
0
5
0
-2
1
-3
0
-1
0
0
1
,
0
0 ,
0
0
0
0
-2
0
0
4
0
0
,
,
0
1
0
0
0
1
0
0
0
0
3 .
-5
0
14)
It can be shown that matrix A is row equivalent to matrix B. Find a basis for Col A.
A)
B)
-1
3
7
2
0
1
-3
-7
1 , -2 , -7 , -1 , 3
0 , 1 , 0
2
-4
-9
-5
1
0
0
5
3
-6
-11
-9
-1
0
0
0
C)
D)
-1
3
7
-1
3
2
1 , -2 , -7
1 , -2 , -1
2
-4
-9
2
-4
-5
3
-6
-11
3
-6
-9
Compute the dot product u ∙ v.
-1
5
15) u = 3 , v = 2
3
-3
A) 0
15)
B) 8
C) -2
4
D) -8