Chapter 4 Vector Spaces- test bank-Introduction to Linear Algebra for Scientists & Engineers

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Introduction to Linear Algebra for Science and Engineering, 2e (Norman & Wolczuk)
Chapter 4 Vector Spaces

4.1 Spaces of Polynomials

Evaluate a linear combination of polynomials

1) Calculate 2(-1 + 2x – 3×2) – 3(5 – 4x – x3).
A) 13 + 16x – 6×2 + 3×3
B) -17 – 8x – 6×2 + 3×3
C) -17 + 16x – 6×2 + 3×3
D) 13 – 8x – 6×2 + 3×3
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (4.1) Spaces of Polynomials
Skill: Applied
Objective: Evaluate a linear combination of polynomials

2) Calculate (-1 + 2x – 3 x2 – x3) – 2(1 – x3).
A) -3 + 2x – 3×2 – 3×3
B) 1 + 2x – 3×2 + x3
C) 3 + 2x – 3×2 + 2×3
D) -3 + 2x – 3×2 + x3
Answer: D
Diff: 1 Type: BI Var: 1
Topic: (4.1) Spaces of Polynomials
Skill: Applied
Objective: Evaluate a linear combination of polynomials

Determine if a polynomial is in the span of a set of polynomials

1) Determine which polynomials are in the span of Β = .
p(x) = 1 – 2x + x2, q(x) = 3 + x2
A) Both p(x) and q(x)
B) q(x) only
C) p(x) only
D) Neither p(x) nor q(x)
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (4.1) Spaces of Polynomials
Skill: Applied
Objective: Determine if a polynomial is in the span of a set of polynomials

2) Determine which polynomials are in the span of Β = .
p(x) = 1 – x + x2, q(x) = 3 + 3x + x2
A) Both p(x) and q(x)
B) q(x) only
C) p(x) only
D) Neither p(x) nor q(x)
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (4.1) Spaces of Polynomials
Skill: Applied
Objective: Determine if a polynomial is in the span of a set of polynomials

3) Determine which polynomials are in the span of Β = .
p(x)=2, q(x)= 3 + x3
A) Both p(x) and q(x)
B) q(x) only
C) p(x) only
D) Neither p(x) nor q(x)
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (4.1) Spaces of Polynomials
Skill: Applied
Objective: Determine if a polynomial is in the span of a set of polynomials

4) Determine which polynomials are in the span of Β = .
p(x)=3 – x2 – 2×3, q(x)= 3 + x3
A) Both p(x) and q(x)
B) q(x) only
C) p(x) only
D) Neither p(x) nor q(x)
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (4.1) Spaces of Polynomials
Skill: Applied
Objective: Determine if a polynomial is in the span of a set of polynomials

Determine if a set of polynomials is linearly independent

Determine which of the sets of vectors is linearly independent.
1) A: The set where P1 (t) = 1, P2 (t) = t2, P3 (t) = 2 + 3t
B: The set where P1 (t) = t, P2 (t) = t2, P3 (t) = 2t + 3t2
C: The set where P1 (t) = 1, P2 (t) = t2, P3 (t) = 2 + 3t + t2
A) A and C
B) A only
C) B only
D) C only
E) All of them
Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (4.1) Spaces of Polynomials
Skill: Applied
Objective: Determine if a set of polynomials is linearly independent

2) A = , B =
A) B only
B) A only
C) Both A and B
D) Neither A nor B
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (4.1) Spaces of Polynomials
Skill: Applied
Objective: Determine if a set of polynomials is linearly independent

3) A = , B =
A) B only
B) A only
C) Both A and B
D) Neither A nor B
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (4.1) Spaces of Polynomials
Skill: Applied
Objective: Determine if a set of polynomials is linearly independent

4.2 Vector Spaces

Determine if a set is a subspace of a vector space

1) 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 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) A only
C) C only
D) Both A and B
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (4.2) Vector Spaces
Skill: Applied
Objective: Determine if a set is a subspace of a vector space

2) Let H be the set of all polynomials having degree at most 4 and rational coefficients. Determine whether H is a subspace of Ρ4. If it is not a subspace, 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 a subspace.
B) H is not a subspace; not closed under multiplication by scalars.
C) H is not a subspace; does not contain zero vector.
D) H is not a subspace; not closed under vector addition.
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (4.2) Vector Spaces
Skill: Applied
Objective: Determine if a set is a subspace of a vector space

3) 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 subspace of Ρ2. If it is not a subspace, 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 subspace; does not contain zero vector.
B) H is not a subspace; not closed under multiplication by scalars.
C) H is not a subspace; not closed under multiplication by scalars and does not contain zero vector.
D) H is not subspace; not closed under vector addition.
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (4.2) Vector Spaces
Skill: Applied
Objective: Determine if a set is a subspace of a vector space
4) Let H be the set of all points of the form (s, s -1). Determine whether H is a subspace of ℛ2. If it is not a subspace, 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 a subspace.
B) H is not a subspace; fails to satisfy all three properties.
C) H is not a subspace; does not contain zero vector.
D) H is not a subspace; not closed under vector addition.
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (4.2) Vector Spaces
Skill: Applied
Objective: Determine if a set is a subspace of a vector space

5) Let W = . Determine whether W is a subspace of M(2,2). If it is not a subspace, 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 a vector space.
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.
D) H is not a vector space; not closed under vector addition.
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (4.2) Vector Spaces
Skill: Applied
Objective: Determine if a set is a subspace of a vector space

6) Let W = . Determine whether W is a subspace of M(2, 2). If it is not a subspace, 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 a subspace.
B) H is not a subspace; fails to satisfy all three properties.
C) H is not a subspace; does not contain zero vector.
D) H is not a subspace; not closed under vector addition and does not contain zero vector.
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (4.2) Vector Spaces
Skill: Applied
Objective: Determine if a set is a subspace of a vector space
4.3 Bases and Dimensions

Determine if a set is a basis for a vector space

Determine which of the sets of vectors is linearly independent.
1) A: The set in C[0, 1]
B: The set in C[0, 1]
C: The set in C[0, 1]
A) A and B
B) A only
C) B only
D) C only
E) A and C
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (4.3) Bases and Dimensions
Skill: Applied
Objective: Determine if a set is a basis for a vector space

2) Determine which of the following sets are a basis for M(2, 2).

B = , C =

A) Neither B nor C
B) C only
C) Both B and C
D) B only
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (4.3) Bases and Dimensions
Skill: Applied
Objective: Determine if a set is a basis for a vector space

3) Determine which of the following sets are a basis for ℛ3.

B = , C =

A) Neither B nor C
B) C only
C) Both B and C
D) B only
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (4.3) Bases and Dimensions
Skill: Applied
Objective: Determine if a set is a basis for a vector space
4) Determine which of the following sets are a bsis for Ρ3.

B = , C =

A) Neither B nor C
B) C only
C) Both B and C
D) B only
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (4.3) Bases and Dimensions
Skill: Applied
Objective: Determine if a set is a basis for a vector space

Determine whether the set of vectors is a basis for ℛ3.
5) Given the set of vectors , decide which of the following statements is true:
A: Set is linearly independent and spans ℛ3. Set is a basis for ℛ3.
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) B
C) C
D) D
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (4.3) Bases and Dimensions
Skill: Applied
Objective: Determine if a set is a basis for a vector space

6) Given the set of vectors , decide which of the following statements is true:
A: Set is linearly independent and spans ℛ3. Set is a basis for ℛ3.
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) B
C) C
D) D
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (4.3) Bases and Dimensions
Skill: Applied
Objective: Determine if a set is a basis for a vector space

Find a basis and dimension of a vector space

1) Let H = .
Find the dimension of the subspace H.
A) dim H = 3
B) dim H = 2
C) dim H = 1
D) dim H = 4
Answer: B
Diff: 3 Type: BI Var: 8
Topic: (4.3) Bases and Dimensions
Skill: Applied
Objective: Find a basis and dimension of a vector space

2) Determine which of the following statements is false.
A: The dimension of the vector space of polynomials is 8.
B: Any line in ℛ3 is a one-dimensional subspace of ℛ3.
C: If a vector space V has a basis B = , then any set in V containing 4 vectors must be linearly dependent.
A) A
B) B
C) C
D) A and B
Answer: B
Diff: 3 Type: BI Var: 25
Topic: (4.3) Bases and Dimensions
Skill: Applied
Objective: Find a basis and dimension of a vector space

3) 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 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.
A) A
B) B
C) C
D) A and C
Answer: C
Diff: 3 Type: BI Var: 25
Topic: (4.3) Bases and Dimensions
Skill: Applied
Objective: Find a basis and dimension of a vector space

4) Let H = .
Find the dimension of the subspace H.
A) dim H = 3
B) dim H = 2
C) dim H = 1
D) dim H = 4
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (4.3) Bases and Dimensions
Skill: Applied
Objective: Find a basis and dimension of a vector space

5) Let H = .
Find the dimension of the subspace H.
A) dim H = 3
B) dim H = 2
C) dim H = 1
D) dim H = 4
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (4.3) Bases and Dimensions
Skill: Applied
Objective: Find a basis and dimension of a vector space

Extending a Linearly Independent Subset to a Basis

1) Let C= . Extend C to a basis for M(2, 2).

Answer: C
Diff: 3 Type: BI Var: 1
Topic: (4.3) Bases and Dimensions
Skill: Applied
Objective: Extending a linearly independent subset to a basis

2) Extend a basis for the plane -2x – 14y + 8z = 0 to obtain a basis for ℛ3.

Answer: B
Diff: 3 Type: BI Var: 1
Topic: (4.3) Bases and Dimensions
Skill: Applied
Objective: Extending a linearly independent subset to a basis

4.4 Coordinates with Respect to a Basis

Find the coordinates of a vector with respect to a basis

Find the vector x determined by the given coordinate vector [x]B and the given basis B.
1) B = , [x]B =

Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (4.4) Coordinates with Respect to a Basis
Skill: Recall
Objective: Find the coordinates of a vector with respect to a basis
Find the coordinate vector [x]B of the vector x relative to the given basis B.
2) b1 = , b2 = , x = , and B =

Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (4.4) Coordinates with Respect to a Basis
Skill: Applied
Objective: Find the coordinates of a vector with respect to a basis

3) b1 = , b2 = , x = , and B =

Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (4.4) Coordinates with Respect to a Basis
Skill: Applied
Objective: Find the coordinates of a vector with respect to a basis
4) B = and = -2 + 4x + 2×2

Answer: C
Diff: 2 Type: BI Var: 1
Topic: (4.4) Coordinates with Respect to a Basis
Skill: Applied
Objective: Find the coordinates of a vector with respect to a basis

5) B = , =

Answer: D
Diff: 2 Type: BI Var: 1
Topic: (4.4) Coordinates with Respect to a Basis
Skill: Applied
Objective: Find the coordinates of a vector with respect to a basis

Find a vector give the coordinates of the vector with respect to a basis

Find the new coordinate vector for the vector x after performing the specified change of basis.
1) Consider two bases B = {b1, b2} and C = {c1, c2} for a vector space V such that
Suppose x = b1 + 2b2. That is, suppose [x]B = . Find [x]C.
A)

Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (4.4) Coordinates with Respect to a Basis
Skill: Applied
Objective: Find a vector give the coordinates of the vector with respect to a basis

2) Consider two bases B = {b1, b2, b3} and C = {c1, c2, c3} for a vector space V such that
That is, suppose Find [x]C

Answer: B
Diff: 3 Type: BI Var: 50+
Topic: (4.4) Coordinates with Respect to a Basis
Skill: Applied
Objective: Find a vector give the coordinates of the vector with respect to a basis

Find a change of coordinates matrix

Find the specified change-of-coordinates matrix.
1) Let B = {b1, b2} and C = {c1, c2} be bases for ℛ2, where

b1 = , b2 = , c1 = , c2 = .

Find the change-of-coordinates matrix from B to C.

Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (4.4) Coordinates with Respect to a Basis
Skill: Applied
Objective: Find a change of coordinates matrix

2) Consider two bases B = {b1, b2} and C = {c1, c2} for a vector space V such that Find the change-of-coordinates matrix from B to C.

Answer: C
Diff: 2 Type: BI Var: 50+
Topic: (4.4) Coordinates with Respect to a Basis
Skill: Applied
Objective: Find a change of coordinates matrix

4.5 General Linear Mappings

Determine with a proof if a mapping is linear

1) Determine which of the following mappings are linear.
ℒ: ℛ2 → P1 defined by ℒ = (a + b) + bx.
Τ: M(2,2) → ℛ defined by Τ = a + d.
A) Neither ℒ nor T
B) Τonly
C) ℒ only
D) Both ℒ and Τ
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (4.6) Matrix of a Linear Mapping
Skill: Applied
Objective: Determine with a proof if a mapping is linear

2) Determine which of the following mappings are linear.
ℒ: ℛ2 → M(2,2) defined by ℒ = .
Τ: M(2,2) → P3 defined by Τ = cx + (a + b) x3
A) Neither ℒ nor Τ
B) Τonly
C) ℒ only
D) Both ℒ and Τ
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (4.6) Matrix of a Linear Mapping
Skill: Applied
Objective: Determine with a proof if a mapping is linear

Determine if a vector is in the range of a linear mapping

1) Let Τ: Ρ2 → M(2, 2) be a linear mapping defined by Τ = .
Determine whether the following matrices are in the range of Τ.
A = and B =
A) Both A and B
B) Neither A nor B
C) A only
D) B only
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (4.5) General Linear Mappings
Skill: Applied
Objective: Determine if a vector is in the range of a linear mapping

2) Let Τ: M(2, 2) → M(2, 2) be linear mapping defined by Τ = .
Determine whether the following matrices are in the range of Τ.

A = and B =
A) Both A and B
B) Neither A nor B
C) A only
D) B only
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (4.5) General Linear Mappings
Skill: Applied
Objective: Determine if a vector is in the range of a linear mapping

Find a basis for the range and kernel of a linear mapping

1) Find a basis for the null space of the linear mapping Τ: M(2, 2) → M(2, 2) defined by
Τ = .
A)
B)
C)
D)
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (4.5) General Linear Mappings
Skill: Applied
Objective: Find a basis for the range and kernel of a linear mapping

2) Find a basis for the range of the linear mapping Τ: M(2, 2) → M(2, 2) defined by
Τ = .
A)
B)
C)
D)
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (4.5) General Linear Mappings
Skill: Applied
Objective: Find a basis for the range and kernel of a linear mapping

3) Find a basis for the null space of the linear mapping Τ: M(2, 2) → ℛ defined by Τ = a + d.
A)
B)
C)
D)
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (4.5) General Linear Mappings
Skill: Applied
Objective: Find a basis for the range and kernel of a linear mapping

4) Find a basis for the range of the linear mapping Τ: M(2, 2) → ℛ defined by Τ = a + d.
A) no basis
B)
C)
D)
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (4.5) General Linear Mappings
Skill: Applied
Objective: Find a basis for the range and kernel of a linear mapping

4.6 Matrix of a Linear Mapping

Determine the matrix of a linear mapping with respect to a given basis B

Find the B- matrix of the linear operator L: V → V relative to B.
1) Let ℒ : ℛ3 → ℛ3 be a linear mapping defined by ℒ( ) = , ℒ( ) = , ℒ( ) = 5 and
B = .

Answer: D
Diff: 2 Type: BI Var: 1
Topic: (4.6) Matrix of a Linear Mapping
Skill: Applied
Objective: Determine the matrix of a linear mapping with respect to a given basis B

2) Let ℒ : ℛ3 → ℛ3 be a linear mapping defined by ℒ(1,0,1) = (1,2,1), ℒ(1,-1,0) = (0,1,-1), and ℒ(0,1,-1) = (3,-3, 0), where
B = .

Answer: C
Diff: 2 Type: BI Var: 1
Topic: (4.6) Matrix of a Linear Mapping
Skill: Applied
Objective: Determine the matrix of a linear mapping with respect to a given basis B

Find the matrix of a linear mapping given its standard matrix

1) Let = . Given a natural basis B = , determine B-matrix of the transformation proj .
A) [proj ] =

B) [proj ] =

C) [proj ] =

D) [proj ] =

Answer: C
Diff: 3 Type: BI Var: 1
Topic: (4.6) Matrix of a Linear Mapping
Skill: Applied
Objective: Find the matrix of a linear mapping given its standard matrix

2) Let ℒ: ℛ3 → ℛ3 be a linear mapping with the standard matrix .
Determine matrix of ℒ with respect to the basis B = .
A)

Answer: D
Diff: 3 Type: BI Var: 1
Topic: (4.6) Matrix of a Linear Mapping
Skill: Applied
Objective: Find the matrix of a linear mapping given its standard matrix

3) Let ℒ : ℛ2 → ℛ2 be a linear mapping with the standard matrix A = . Determine matrix of ℒ with respect to the basis B = .
A)

Answer: B
Diff: 3 Type: BI Var: 1
Topic: (4.6) Matrix of a Linear Mapping
Skill: Applied
Objective: Find the matrix of a linear mapping given its standard matrix

4) Let ℒ: ℛ2 → ℛ2 be a linear mapping with the standard matrix A = . Determine matrix of ℒ with respect to the basis B = .
A)

Answer: D
Diff: 3 Type: BI Var: 1
Topic: (4.6) Matrix of a Linear Mapping
Skill: Applied
Objective: Find the matrix of a linear mapping given its standard matrix

Use a matrix of a linear mapping to find the image of a vector under the mapping

1) Let ℒ: ℛ3 → ℛ3 be a linear mapping with the B-matrix , where basis B = .
Determine ℒ(0.1,0).
A) (-1, 4, -1)
B) (4, 1, -1)
C) (3, 1, -2)
D) (1, 4, 1)
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (4.6) Matrix of a Linear Mapping
Skill: Applied
Objective: Use a matrix of a linear mapping to find the image of a vector under the mapping

2) Let ℒ: ℛ2 → ℛ2 be a linear mapping with the B-matrix , where basis B = . Determine ℒ(0,1).
A) (3, -5)
B) (3, 5)
C) (2, 4)
D) (-2, 4)
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (4.7) Isomorphisms of Vector Spaces
Skill: Recall
Objective: Use a matrix of a linear mapping to find the image of a vector under the mapping

Find the matrix of a linear mapping with respect to bases B and C

Find the matrix of the linear mapping T: V → W relative to B and C.
1) Suppose B = {b1, b2} is a basis for V, and C = {c1, c2, c3} is a basis for W. Let T be defined by
T(b1) = -5c1 – 6c2 + 5c3
T(b2) = -5c1 – 12c2 + 2c3

Answer: A
Diff: 1 Type: BI Var: 50+
Topic: (4.6) Matrix of a Linear Mapping
Skill: Applied
Objective: Find the matrix of a linear mapping with respect to bases B and C

2) Suppose B = {b1, b2} is a basis for V, and C = {c1, c2, c3} is a basis for W. Let T be defined by

T(b1) = -5c1 – 6c2 + 5c3
T(b2) = -5c1 – 12c2 + 2c3

Answer: A
Diff: 1 Type: BI Var: 50+
Topic: (4.6) Matrix of a Linear Mapping
Skill: Applied
Objective: Find the matrix of a linear mapping with respect to bases B and C
4.7 Isomorphisms of Vector Spaces

Determine if two subspaces are isomorphic

1) Determine which of the following pairs of vector spaces are isomorphic.
I) P = and Γ = .
II) P2 and ℛ2
A) Neither I nor II
B) II only
C) I only
D) Both I and II
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (4.7) Isomorphisms of Vector Spaces
Skill: Applied
Objective: Determine if two subspaces are isomorphic

2) Which of the following statements are true?
I) If the vector spaces U and V have the same finite dimension then they are isormorphic.
II) Any plane through the origin in ℛ3 is isormorphic to ℛ2.
A) I only
B) Both I and II
C) II only
D) Neither I nor II
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (4.7) Isomorphisms of Vector Spaces
Skill: Applied
Objective: Determine if two subspaces are isomorphic

Demonstrate knowledge of one-to-one and onto mappings

Determine whether the linear transformation T is one-to-one and whether it maps as specified.
1) Let T be the linear transformation whose standard matrix is
A = .
Determine whether the linear transformation T is one-to-one and whether it maps ℛ2 onto ℛ3.
A) One-to-one; onto ℛ3
B) One-to-one; not onto ℛ3
C) Not one-to-one; onto ℛ3
D) Not one-to-one; not onto ℛ3
Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (4.7) Isomorphisms of Vector Spaces
Skill: Applied
Objective: Demonstrate knowledge of one-to-one and onto mappings
2) T(x1, x2, x3) = (-2×2 – 2×3, -2×1 + 8×2 + 4×3, – x1 – 2×3, 4×2 + 4×3)
Determine whether the linear transformation T is one-to-one and whether it maps ℛ3 onto ℛ4.
A) Not one-to-one; onto ℛ4
B) One-to-one; not onto ℛ4
C) Not one-to-one; not onto ℛ4
D) One-to-one; onto ℛ4
Answer: C
Diff: 3 Type: BI Var: 36
Topic: (4.7) Isomorphisms of Vector Spaces
Skill: Applied
Objective: Demonstrate knowledge of one-to-one and onto mappings