Chapter 1 Euclidean Vector Spaces- test bank-Introduction to Linear Algebra for Scientists & Engineers

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Chapter 1 Euclidean Vector Spaces- test bank-Introduction to Linear Algebra for Scientists & Engineers
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Introduction to Linear Algebra for Scientists & Engineers, 2e (Norman & Wolczuk)
Chapter 1 Euclidean Vector Spaces

1.1 Vectors in R2 and R3

Evaluate linear combinations of vectors in R2 and R3

1) Let = , = . Find + .
A)

2) Let u = , v = . Find u – v.
A)

3) Let u = , v = . Find v – u.
A)

4) Let u = . Find 6u.
A)

5) Let u = . Find 7u.
A)

6) Let u = . Find -9u.
A)

7) Let u = . Find -5u.
A)

8) Let u = , v = . Find 2u + v.
A)

9) Let u = and v = . Display the vectors u, v, and u + v on the same axes.

10) Let u = . Display the vector 2u using the given axes.

1) Consider the points P(1,2,2), Q(-1, 0, 1), and R(3, 2, 1). Then
A) + =
B) + =
C) + =
D) + =

The Vector Equation of a Line in R2 and R3

1) Find a vector equation of the line passing through the point P(1, -1, 3) and parallel to the line with an equation = + t , t ∈ ℛ.
A) = + t , t ∈ ℛ.
B) = + t , t ∈ ℛ.
C) = + t , t ∈ ℛ.
D) = + t , t ∈ ℛ.

2) Find a vector equation of the line passing through the points P(-2, 1, 3) and Q(3, 0, -2).
A) = + t , t ∈ ℛ.
B) = + t , t ∈ ℛ.
C) = + t , t ∈ ℛ.
D) = + t , t ∈ ℛ.

3) Find a vector equation of the line passing through the points P(3, -1, 4) and Q(5, 1, 5).
A) = + t , t ∈ ℛ.
B) = + t , t ∈ ℛ.
C) = + t , t ∈ ℛ.
D) = + t , t ∈ ℛ.
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (1.1) Vectors in R2 and R3
Skill: Applied
Objective: The vector equation of a line in R2 and R3
1.2 Vectors in Rn

Evaluate linear combinations of vectors in Rn

1) Let a1 = , a2 = , and b = .
Determine whether b can be written as a linear combination of a1 and a2. In other words, determine whether weights x1 and x2 exist, such that x1 a1 + x2 a2 = b. Determine the weights x1 and x2 if possible.
A) x1 = -3, x2 = 2
B) x1 = -2, x2 = 1
C) x1 = -3, x2 = 3
D) No solution
Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (1.2) Vectors in Rn
Skill: Applied
Objective: Evaluate linear combinations of vectors in Rn

2) Let a1 = , a2 = , a3 = , and b = .
Determine whether b can be written as a linear combination of a1, a2, and a3. In other words, determine whether weights x1, x2, and x3 exist, such that x1 a1 + x2 a2 + x3 a3 = b. Determine the weights x1, x2, and , if possible.
A) x1 = -2, x2 = -1, x3 = 2
B) x1 = -6, x2 = 0, x3 = 1
C) x1 = 2, x2 = 1, x3 = –
D) No solution
Answer: D
Diff: 3 Type: BI Var: 50+
Topic: (1.2) Vectors in Rn
Skill: Applied
Objective: Evaluate linear combinations of vectors in Rn
Determine if a set is a subspace of Rn

1) Determine which of the following sets is a subspace of ℛ2.
V is the line y = x in the xy-plane: V =
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) W only
B) U only
C) V only
D) U and V
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (1.2) Vectors in Rn
Skill: Applied
Objective: Determine if a set is a subspace of Rn

Solve the problem.
2) 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; fails to satisfy all three properties.
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; not closed under vector addition.
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (1.2) Vectors in Rn
Skill: Applied
Objective: Determine if a set is a subspace of Rn
3) Let H be the set of all points in the xy-plane having at least one nonzero coordinate: . 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; fails to satisfy all three properties.
B) H is not a vector space; does not contain zero vector and not closed under multiplication by scalars.
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: (1.2) Vectors in Rn
Skill: Applied
Objective: Determine if a set is a subspace of Rn

Determine if a set is linearly independent

1) Determine which of the following sets are linearly independent.
V = , U =

A) V and U
B) U only
C) V only
D) None of them
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (1.2) Vectors in Rn
Skill: Applied
Objective: Determine if a set is linearly independent

2) For what values of h are the given vectors linearly independent?
,
A) Vectors are linearly independent for h ≠ -4.
B) Vectors are linearly independent for h = -4.
C) Vectors are linearly independent for all h.
D) Vectors are linearly dependent for all h.
Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (1.2) Vectors in Rn
Skill: Applied
Objective: Determine if a set is linearly independent
3) For what values of h are the given vectors linearly dependent?
, , ,
A) Vectors are linearly independent for all h.
B) Vectors are linearly dependent for h = -24.
C) Vectors are linearly dependent for all h.
D) Vectors are linearly dependent for h ≠ -24.
Answer: C
Diff: 3 Type: BI Var: 50+
Topic: (1.2) Vectors in Rn
Skill: Applied
Objective: Determine if a set is linearly independent

Identify if a set represents a line, plane, or hyperplane

1) Determine if the following set represent a line, plane or hyperplane in ℛ4.
Span .
A) point
B) line
C) plane
D) hyperplane
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (1.2) Vectors in Rn
Skill: Applied
Objective: Identify if a set represents a line, plane, or hyperplane

1.3 Length and Dot Products

Calculate dot products and length in Rn

Compute the dot product u ∙ v.
1) u = , v =
A) 210
B) 246
C) 174
D) -36
Answer: C
Diff: 1 Type: BI Var: 50+
Topic: (1.3) Length and Dot Products
Skill: Recall
Objective: Calculate dot products and lengths in Rn
2) u = , v =
A) 72
B) -180
C) 48
D) 60
Answer: D
Diff: 1 Type: BI Var: 50+
Topic: (1.3) Length and Dot Products
Skill: Recall
Objective: Calculate dot products and lengths in Rn

3) u = , v =
A) 21
B) 7
C) 23
D) 100
Answer: A
Diff: 1 Type: BI Var: 50+
Topic: (1.3) Length and Dot Products
Skill: Recall
Objective: Calculate dot products and lengths in Rn

4) u = , v =
A) 8
B) -8
C) -2
D) 0
Answer: B
Diff: 1 Type: BI Var: 50+
Topic: (1.3) Length and Dot Products
Skill: Recall
Objective: Calculate dot products and lengths in Rn

5) u = , v =
A) 0
B) -21
C) -40
D) -24
Answer: D
Diff: 1 Type: BI Var: 50+
Topic: (1.3) Length and Dot Products
Skill: Recall
Objective: Calculate dot products and lengths in Rn

Find a unit vector in the direction of the given vector.
6)
A)

Answer: A
Diff: 1 Type: BI Var: 28
Topic: (1.3) Length and Dot Products
Skill: Recall
Objective: Calculate dot products and lengths in Rn

D)
Answer: C
Diff: 1 Type: BI Var: 18
Topic: (1.3) Length and Dot Products
Skill: Recall
Objective: Calculate dot products and lengths in Rn

Find the distance between the two points.
8) u = (6, -1), v = (1, 6)
A)
B) 74
C)
D) 37
Answer: A
Diff: 1 Type: BI Var: 10
Topic: (1.3) Length and Dot Products
Skill: Applied
Objective: Calculate dot products and lengths in Rn

9) u = (-8, 16), v = (16, -16)
A) 8
B) 1,600
C) 40
D) 200
Answer: C
Diff: 1 Type: BI Var: 16
Topic: (1.3) Length and Dot Products
Skill: Applied
Objective: Calculate dot products and lengths in Rn
10) u = (0, 0, 0), v = (6, 9, 9)
A) 3
B) 2
C) 24
D) 198
Answer: A
Diff: 1 Type: BI Var: 31
Topic: (1.3) Length and Dot Products
Skill: Applied
Objective: Calculate dot products and lengths in Rn

11) u = (0, 0, 0), v = (-8, -6, -2)
A) 2
B) 2
C) -16
D) 104
Answer: B
Diff: 1 Type: BI Var: 50+
Topic: (1.3) Length and Dot Products
Skill: Applied
Objective: Calculate dot products and lengths in Rn

12) u = (-8, 2, -6), v = (-3, 3, 4)
A) 16
B) 5
C) 3
D) 126
Answer: C
Diff: 1 Type: BI Var: 50+
Topic: (1.3) Length and Dot Products
Skill: Applied
Objective: Calculate dot products and lengths in Rn

13) u = (14, 17, 20), v = (1, 7, 4)
A) 525
B) 9
C) 39
D) 5
Answer: D
Diff: 1 Type: BI Var: 50+
Topic: (1.3) Length and Dot Products
Skill: Applied
Objective: Calculate dot products and lengths in Rn
14) u = (25, 16, 21), v = (-3, 2, 7)
A) 14
B) 2
C) 14
D) 1,176
Answer: C
Diff: 1 Type: BI Var: 50+
Topic: (1.3) Length and Dot Products
Skill: Applied
Objective: Calculate dot products and lengths in Rn

Determine if two vectors are orthogonal

1) Determine which of the following set of vectors are orthogonal.
V = , U =
A) Neither
B) U only
C) U and V
D) V only
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (1.3) Length and Dot Products
Skill: Applied
Objective: Determine if two vectors are orthogonal

2) Determine for what values of k the pair of vectors is orthogonal.
A) 0
B)
C) k = ±
D) –
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (1.3) Length and Dot Products
Skill: Applied
Objective: Determine if two vectors are orthogonal

3) Find the angle in ℛ3 between = and = .
A) π/4
B) π/6
C) π/3
D) π/2
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (1.3) Length and Dot Products
Skill: Applied
Objective: Determine if two vectors are orthogonal
Find a Scalar Equation and normal vector of Planes and Hyperplanes

1) Determine a normal vector of the hyperplane 3×1 + 2×2 -4×3 = 5 in ℛ4.
A)

Answer: C
Diff: 1 Type: BI Var: 1
Topic: (1.3) Length and Dot Products
Skill: Recall
Objective: Find a scalar equation and normal vector of planes and hyperplanes

2) Find an equation for the plane through the point P(-1, 2, 4) and parallel to the plane 2x + y – 2z = 4.
A) 2x + y – 2z = -4
B) 2x + y – 2z = 4
C) 2x + y – 2z = 8
D) 2x + y – 2z + 8 = 0
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (1.3) Length and Dot Products
Skill: Applied
Objective: Find a scalar equation and normal vector of planes and hyperplanes

3) Find an equation of the plane such that each point of the plane is equidistant from the points P(1, 2, 1) and Q(-1, 0, 3).
A) x + 2y – z =- 1
B) x + y – z = 3
C) x + y – z = -1
D) -x + y – z = -1
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (1.3) Length and Dot Products
Skill: Applied
Objective: Find a scalar equation and normal vector of planes and hyperplanes
1.4 Projections and Minimum Distance

Find the projection and perpendicular of a projection of vectors in Rn

Find the orthogonal projection of onto .
1) = , =

Answer: B
Diff: 2 Type: BI Var: 32
Topic: (1.4) Projections and Minimum Distance
Skill: Applied
Objective: Find the projection and perpendicular of a projection of vectors in Rn

2) = , =

Answer: A
Diff: 2 Type: BI Var: 20
Topic: (1.4) Projections and Minimum Distance
Skill: Applied
Objective: Find the projection and perpendicular of a projection of vectors in Rn

3) Let = and = . Then find the projection of onto .
A)

Answer: D
Diff: 2 Type: BI Var: 1
Topic: (1.4) Projections and Minimum Distance
Skill: Recall
Objective: Find the projection and perpendicular of a projection of vectors in Rn

Find a point on a line closest to another point

1) Find the point on the line = + t , t ∈ℛ, closest to Q(0, 0, 1).
A)
B)
C)
D)
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (1.4) Projections and Minimum Distance
Skill: Applied
Objective: Find a point on a line closest to another point

Find the distance from a point to a line/plane/hyperplane

1) Find the distance from the point Q(0,0,1) to the line = + t , t ∈ ℛ.
A)
B)
C)
D)
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (1.4) Projections and Minimum Distance
Skill: Applied
Objective: Find the distance from a point to a line/plane/hyperplane

2) Find the distance from the point Q(1,-1, 2) to the plane x + y + z = 1.
A)
B)
C)
D)
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (1.4) Projections and Minimum Distance
Skill: Applied
Objective: Find the distance from a point to a line/plane/hyperplane

1.5 Cross-Products and Volumes

Calculate the cross product of two vectors in R3

Calculate the cross product of two vectors in ℛ3.
1) ×
A)

Answer: B
Diff: 1 Type: BI Var: 1
Topic: (1.5) Cross-Products and Volumes
Skill: Recall
Objective: Calculate the cross product of two vectors in R3

2) ×

Answer: C
Diff: 1 Type: BI Var: 1
Topic: (1.5) Cross-Products and Volumes
Skill: Recall
Objective: Calculate the cross product of two vectors in R3

3) Let and be vectors in ℛ3. Then ( × ) =
A)
B) 0
C)
D)
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (1.5) Cross-Products and Volumes
Skill: Applied
Objective: Calculate the cross product of two vectors in R3

4) Find the distance between the lines = + t , t ∈ ℛ, and = + s , s ∈ ℛ.
A) 2
B) 3
C) 1
D) 5
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (1.5) Cross-Products and Volumes
Skill: Applied
Objective: Calculate the cross product of two vectors in R3

Finding the normal vector of a plane

1) Find the normal vector of the plane that contains (1,1,1), (3,-1,5), and (4,2,-2).
A) 8×1 + 2×2 + 4×3 = 14
B) -2×1 + 9×2 + x3 = 8
C) x1 + x2 + x3 =
D) x1 + 9×2 + 4×3 = 14
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (1.5) Cross-Products and Volumes
Skill: Applied
Objective: Finding the normal vector of a plane

2) Determine a vector equation of the line of intersection of the planes x + y+ z = 3 and x – y + z = 2.
A) = + t , t ∈ ℛ
B) = + t , t ∈ ℛ
C) = + t , t ∈ ℛ
D) = + t , t ∈ ℛ
Answer: D
Diff: 1 Type: BI Var: 1
Topic: (1.5) Cross-Products and Volumes
Skill: Applied
Objective: Finding the normal vector of a plane

Use the cross product to find the area of a parallelogram

1) Calculate the area of the parallelogram induced by = , = .
A) -1
B) 1
C) 7
D) -3
Answer: B
Diff: 1 Type: BI Var: 1
Topic: (1.5) Cross-Products and Volumes
Skill: Applied
Objective: Use the cross product to find the area of a parallelogram

Use the cross product to find the area of a parallelogram.
2) (0, 0), (5, 8), (13, 10), (8, 2)
A) 59
B) 108
C) 54
D) 53
Answer: C
Diff: 2 Type: BI Var: 50+
Topic: (1.5) Cross-Products and Volumes
Skill: Applied
Objective: Use the cross product to find the area of a parallelogram

3) (-1, -2), (3, 6), (5, 0), (9, 8)
A) 40
B) 80
C) 44
D) 39
Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (1.5) Cross-Products and Volumes
Skill: Applied
Objective: Use the cross product to find the area of a parallelogram

Find the scalar equation of a plane from a vector equation

1) Determine the scalar equation of the plane with vector equation = + t + s .
A) -4x – 3y + 2z = 6
B) 4x + 3y – 2z = 6
C) -4x – 3y + 2z = -1
D) 4x + 3y – 2z = – 6
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (1.5) Cross-Products and Volumes
Skill: Applied
Objective: Find the scalar equation of a plane from a vector equation

Find the volume of a parallelpiped
1) Compute the volume of the parallepiped determined by , , and .
A) 3
B) -7
C) 7
D) 5
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (1.5) Cross-Products and Volumes
Skill: Applied
Objective: Find the volume of a parallelpiped