Chapter 2 Systems of Linear Equations- 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 2 Systems of Linear Equations

2.1 Systems of Linear Equations and Elimination

Solve a system from REF by back substitution

Solve the system from REF by back substitution.
1) x1 – x2 + 3×3 = -8
x2 + x3 = 0
x3 = 0

A) (8, 8, 0)
B) (-8, 0, 0)
C) (0, -8, -8)
D) (0, 8, -8)
Answer: B
Diff: 1 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system from REF by back substitution

2) x1 – x2 + 3×3 = -8
x2 + x3 = 2

A) (8, 8, 0)
B) No solution
C) (0, -8, -8)
D) = , t ∈ ℛ.

Answer: D
Diff: 1 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system from REF by back substitution

The augmented matrix is given for a system of equations. If the system is consistent, find the general solution. Otherwise state that there is no solution.
3)

A) No solution
B) (-5, 7)
C) = , t ∈ ℛ

D) = , t ∈ ℛ

Answer: A
Diff: 1 Type: BI Var: 50+
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system from REF by back substitution

4)

A) = , t,s ∈ ℛ

B) = , t ∈ ℛ

C) x = -25 + 11z
y = 8 – 4z
z = 6

D) No solution
Answer: D
Diff: 1 Type: BI Var: 50+
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system from REF by back substitution

5)

A) x = 17
y = -6
z = 1

B) x = 5 -2y + 3z
y = -6 – 4z
z is free

C) = , t ∈ ℛ

D) = , t,s ∈ ℛ.

Answer: C
Diff: 1 Type: BI Var: 50+
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system from REF by back substitution

6)

A) = , t ∈ ℛ

B) x = 7 – 6z
y = -2 + 2z
z= 0

C) x = 7 – 6z
y is free
z = 1 + y
D) No solution
Answer: A
Diff: 1 Type: BI Var: 50+
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system from REF by back substitution

7)

A) = , t,s ∈ ℛ

B) x1 = 11 – 4×2 – 3×4
x2 is free
x3 = 5 – 3×4
x4 = 0

C) x1 = 11 – 4×2 – 3×3
x2 = 5 – 3×3
x3 is free

D) x1 = – 4x2x2 + 2×3 + 3×4 + 1
x2 is free
x3 = 5 – 3×4
is free

Answer: A
Diff: 1 Type: BI Var: 50+
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system from REF by back substitution

8)

A) No solution
B) x1 = -6×2 – 6×3 + x4 – 2×5 + 5
x2 is free
x3 is free
x4 = x5 – 1
= -4

C) x1 = -6×2 – 6×3 + 9
x2 is free
x3 = -4
x4 = x5 – 1
x5 = -4

D) x1 = -6×2 – 6×3 + 9
x2 is free
x3 is free
x4 = -4
x5 = -4

Answer: D
Diff: 1 Type: BI Var: 50+
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system from REF by back substitution

Determine if a matrix is in REF

Determine which of the following matrices are in row echelon form.
1) A = , B =

A) Neither
B) B only
C) A only
D) Both A and B
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Determine if a matrix is in REF
2) A = , B =

A) Both A and B
B) B only
C) A only
D) Neither
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Determine if a matrix is in REF

3) A = , B =

A) Both A and B
B) B only
C) A only
D) Neither
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Determine if a matrix is in REF

Row reduce a matrix into REF

1) Row reduce the matrix to obtain a row equivalent matrix in the echelon form.

A)

B)

C)

D)
Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Row reduce a matrix into REF

2) Row reduce the matrix to obtain a row equivalent matrix in the echelon form.

A)

B)

C)

D)
Answer: B
Diff: 2 Type: BI Var: 5
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Row reduce a matrix into REF

Solve a system of linear equations or show that it is inconsistent

Solve the system of linear equations or show that it is inconsistent.
1) x1 – x2 + 3×3 = -8
2×1 + x3 = 0
x1 + 5×2 + x3 = 40
A) (0, 8, 0)
B) (-8, 0, 0)
C) (8, 8, 0)
D) (0, -8, -8)
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system of linear equations or show that it is inconsistent

2) x1 + 3×2 + 2×3 = 11
4×2 + 9×3 = -12
x3 = -4
A) (4, 6, 1)
B) (1, -4, 6)
C) (-4, 1, 6)
D) (1, 6, -4)
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system of linear equations or show that it is inconsistent

3) x1 – x2 + 8×3 = -107
6×1 + x3 = 17
3×2 – 5×3 = 89
A) (5, 8, 13)
B) (5, 8, -13)
C) (5, -8, -13)
D) (-5, -8, 13)
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system of linear equations or show that it is inconsistent

4) 4×1 – x2 + 3×3 = 12
2×1 + 9×3 = -5
x1 + 4×2 + 6×3 = -32
A) (2, -7, -1)
B) (2, 7, -1)
C) (2, -7, 1)
D) (2, 7, 1)
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system of linear equations or show that it is inconsistent

5) x1 + x2 + x3 = 6
x1 – x3 = -2
+ 3×3 = 11
A) (1, -2, 3)
B) (0, 1, 2)
C) (1, 2, 3)
D) No solution
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system of linear equations or show that it is inconsistent
6) x1 + x2 + x3 = 7
x1 – x2 + 2×3 = 7
5×1 + x2 + x3 = 11
A) (1, 2, 4)
B) (1, 4, 2)
C) (4, 2, 1)
D) (4, 1, 2)
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system of linear equations or show that it is inconsistent

7) x1 – x2 + x3 = 8
x1 + x2 + x3 = 6
x1 + x2 – x3 = -12
A) (-2, 1, 9)
B) (-2, -1, 9)
C) (-2, -1, -9)
D) (2, -1, -9)
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system of linear equations or show that it is inconsistent

8) 5×1 + 2×2 + x3 = -11
2×1 – 3×2 – x3 = 17
7×1 + x2 + 2×3 = -4
A) (0, -6, -1)
B) (0, -6, 1)
C) (-3, 0, 4)
D) (3, 0, -4)
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system of linear equations or show that it is inconsistent

9) 7×1 + 7×2 + x3 = 1
x1 + 8×2 + 8×3 = 8
9×1 + x2 + 9×3 = 9
A) (0, 1, 0)
B) (1, -1, 1)
C) (0, 0, 1)
D) (-1, 1, 1)
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system of linear equations or show that it is inconsistent
10) 2×1 + x2 = 0
x1 – 3×2 + x3 = 0
3×1 + x2 – x3 = 0
A) (1, 0, 0)
B) (0, 0, 0)
C) (0, 1, 0)
D) No solution
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system of linear equations or show that it is inconsistent

11) x1 + x2 + x3 = 7
x1 – x2 + 2×3 = 7
2×1 + 3×3 = 15
A) (1, 2, 3)
B) (1, -1, 1)
C) No solution
D) (1, 0, 1)
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system of linear equations or show that it is inconsistent

12) x1 + 3×2 + 2×3 = 11
4×2 + 9×3 = -12
x1 + 7×2 + 11 = -11
A) (0, 0, 0)
B) (1, 0, 0)
C) (0, 1, 0)
D) No solution
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system of linear equations or show that it is inconsistent

13) 5×1 + 2×2 + x3 = -11
2×1 – 3×2 – x3 = 17
7×1 – x2 = 12
A) (1, 0, 1)
B) No solution
C) (1, -2, 1)
D) (1, 5, 1)
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system of linear equations or show that it is inconsistent

14) 3×2 + = -7
x1 + x2 + 2×3 – x4 = 12
3×1 + x3 + 2×4 = 12
x1 + x2 + 5×3 = 26
A) (1, 0, 1, -1)
B) (3, -2, 5, -1)
C) (1, -2, 1, -1)
D) (1, 5, 1, -1)
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system of linear equations or show that it is inconsistent

15) 2×1 – 5×2 + 3×3 = -1
-2×1 + 6×2 – 5×3 = 6
-4×1 + 7×2 = -13
A) = + t

B) = + t

C) = + t

D) = + t
Answer: A
Diff: 2 Type: BI Var: 20
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a system of linear equations or show that it is inconsistent

Solve a word problem by creating and solving a system of linear equations

Solve the word problem by creating and solving a system of linear equations.
1) Suppose an economy consists of three sectors: Energy (E), Manufacturing (M), and Agriculture (A).
Sector E sells 70% of its output to M and 30% to A.
Sector M sells 30% of its output to E, 50% to A, and retains the rest.
Sector A sells 15% of its output to E, 30% to M, and retains the rest.

Denote the prices (dollar values) of the total annual outputs of the Energy, Manufacturing, and Agriculture sectors by pe, pm, and pa, respectively. If possible, find equilibrium prices that make each sector’s income match its expenditures. Find the general solution as a vector, with pa free.
A) =

B) =

C) =

D) =
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a word problem by creating and solving a system of linear equations

2) The network in the figure shows the traffic flow (in vehicles per hour) over several one-way streets in the downtown area of a certain city during a typical lunch time. Determine the general flow pattern for the network. In other words, find the general solution of the system of equations that describes the flow. In your general solution, let x4 be free.

A) x1 = 600 – x4
x2 = 400 – x4
x3 = 300 – x4
x4 is free
x5 = 300
B) x1 = 600 + x5
x2 = 400 – x5
x3 = 300 – x5
x4 = 300
x5 is free
C) x1 = 600 – x4
x2 = 400 + x4
x3 = 300 – x4
x4 is free
x5 = 300
D) x1 = 500 + x4
x2 = 400 – x4
x3 = 300 – x4
x4 is free
x5 = 200
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a word problem by creating and solving a system of linear equations

3) The table shows the amount (in g) of protein, carbohydrate, and fat supplied by one unit (100 g) of three different foods.

Betty would like to prepare a meal using some combination of these three foods. She would like the meal to contain 15 g of protein, 25 g of carbohydrate, and 3 g of fat. How many units of each food should she use so that the meal will contain the desired amounts of protein, carbohydrate, and fat? Round to 3 decimal places.
A) 0.360 units of Food 1, 0.204 units of Food 2, 0.055 units of Food 3
B) 0.302 units of Food 1, 0.238 units of Food 2, 0.085 units of Food 3
C) 0.326 units of Food 1, 0.247 units of Food 2, 0.059 units of Food 3
D) 0.280 units of Food 1, 0.192 units of Food 2, 0.164 units of Food 3
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a word problem by creating and solving a system of linear equations

4) The population of a city in 2000 was 600,000 while the population of the suburbs of that city in 2000 was 900,000.
Suppose that demographic studies show that each year about 5% of the city’s population moves to the suburbs (and 95% stays in the city), while 2% of the suburban population moves to the city (and 98% remains in the suburbs).
Compute the population of the city and of the suburbs in the year 2002. For simplicity, ignore other influences on the population such as births, deaths, and migration into and out of the city/suburban region.
A) City: 576,840
Suburbs: 923,160
B) City: 588,000
Suburbs: 912,000
C) City: 541,500
Suburbs: 864,360
D) City: 541,500
Suburbs: 958,500
Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (2.1) Systems of Linear Equations and Elimination
Skill: Applied
Objective: Solve a word problem by creating and solving a system of linear equations

2.2 Reduced Row Echelon Form, Rank, and Homogeneous Systems

Row reduce a matrix into RREFand determine its rank

Determine the rank of the matrix.
1)
A) 1
B) 3
C) 4
D) 2
Answer: D
Diff: 2 Type: BI Var: 48
Topic: (2.2) Reduced Row Echelon Form, Rank, and Homogeneous Systems
Skill: Applied
Objective: Row reduce a matrix into RREF and determine its rank

2)
A) 2
B) 3
C) 4
D) 5
Answer: B
Diff: 2 Type: BI Var: 50+
Topic: (2.2) Reduced Row Echelon Form, Rank, and Homogeneous Systems
Skill: Applied
Objective: Row reduce a matrix into RREF and determine its rank

3) Row reduce the matrix to obtain a row equivalent matrix in the reduced echelon form.

A)

B)

C)

D)
Answer: A
Diff: 2 Type: BI Var: 5
Topic: (2.2) Reduced Row Echelon Form, Rank, and Homogeneous Systems
Skill: Applied
Objective: Row reduce a matrix into REF

Solve a homogeneous system of linear equations by row reducing to RREF

1) Find the general solution of the simple homogeneous “system” below, which consists of a single linear equation. Give your answer as a linear combination of vectors. Let x2 and x3 be free variables.
-2×1 – 14×2 + 8×3 = 0
A) = x2 + x3 (with x2, x3 free)

B) = x2 + x3 (with x2, x3 free)

C) = x2 + x3 (with x2, x3 free)

D) = -7 – 4 (with x2, x3 free)
Answer: B
Diff: 2 Type: BI Var: 50+
Topic: (2.2) Reduced Row Echelon Form, Rank, and Homogeneous Systems
Skill: Applied
Objective: Solve a homogeneous system of linear equations by row reducing to RREF

2) Find the general solution of the homogeneous system below. Give your answer as a vector.
x1 + 2×2 – 3×3 = 0
4×1 + 7×2 – 9×3 = 0
-x1 – 3×2 + 6×3 = 0
A) = x3

B) = x3

C) =

D) = x3
Answer: A
Diff: 2 Type: BI Var: 4
Topic: (2.2) Reduced Row Echelon Form, Rank, and Homogeneous Systems
Skill: Applied
Objective: Solve a homogeneous system of linear equations by row reducing to RREF

3) Describe all solutions of Ax = b, where
A = and = .

Describe the general solution in parametric vector form.
A) = + x3

B) = + x3

C) = + x3

D) = + x3
Answer: D
Diff: 3 Type: BI Var: 20
Topic: (2.2) Reduced Row Echelon Form, Rank, and Homogeneous Systems
Skill: Applied
Objective: Solve a homogeneous system of linear equations by row reducing to RREF

Demonstrate understanding of theory of solving systems

1) Let A = and = .

Determine if the equation Ax = b is consistent for all possible b1, b2, b3. If the equation is not consistent for all possible b1, b2, b3, give a description of the set of all b for which the equation is consistent (i.e., a condition which must be satisfied by b1, b2, b3)
A) Equation is consistent for all possible b1, b2, b3.
B) Equation is consistent for all b1, b2, b3 satisfying 7b1 + 5b2 + b3 = 0.
C) Equation is consistent for all b1, b2, b3 satisfying -3b1 + b3 = 0.
D) Equation is consistent for all b1, b2, b3 satisfying 2b1 + b2 = 0.
Answer: A
Diff: 3 Type: BI Var: 16
Topic: (2.2) Reduced Row Echelon Form, Rank, and Homogeneous Systems
Skill: Applied
Objective: Demonstrate understanding of theory of solving systems

2) Let A = and b = .

Determine if the equation Ax = b is consistent for all possible b1, b2, b3. If the equation is not consistent for all possible b1, b2, b3, give a description of the set of all b for which the equation is consistent (i.e., a condition which must be satisfied by b1, b2, b3).
A) Equation is consistent for all possible b1, b2, b3.
B) Equation is consistent for all b1, b2, b3 satisfying -b1 + b2 + b3 = 0.
C) Equation is consistent for all b1, b2, b3 satisfying -3b1 + b3 = 0.
D) Equation is consistent for all b1, b2, b3 satisfying 3b1 + 3b2 + b3 = 0.
Answer: D
Diff: 3 Type: BI Var: 4
Topic: (2.2) Reduced Row Echelon Form, Rank, and Homogeneous Systems
Skill: Applied
Objective: Demonstrate understanding of theory of solving systems

3) Find the conditions on a,b such that the system

is inconsistent.
A) a = 2 and b = 1
B) a ≠ 2 and b ≠ 1
C) a = 2 and b ≠ 1
D) a ≠ 2 and b = 1
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (2.2) Reduced Row Echelon Form, Rank, and Homogeneous Systems
Skill: Applied
Objective: Demonstrate understanding of theory of solving systems

4) Find the conditions on a such that the system

has a unique solution.
A) a = 0 or a = 1
B) a ≠ 0 and a ≠ -1
C) a = 0
D) a = -1
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (2.2) Reduced Row Echelon Form, Rank, and Homogeneous Systems
Skill: Applied
Objective: Demonstrate understanding of theory of solving systems

2.3 Application to Spanning and Linear Independence

Determine if a vector is in the span of a set

Determine if the vector p is in Span S.
1) Consider the set S = .
Let p = and q = .
Then
A) p ∈ Span S and q ∈ Span S .
B) p ∉ Span S and q ∉ Span S.
C) p ∉ Span S and q ∈ Span S.
D) p ∈ Span S and q ∉ Span S.
Answer: C
Diff: 3 Type: MC Var: 1
Topic: (2.3) Application to Spanning and Linear Independence
Skill: Applied
Objective: Determine if a vector is in the span of a set

Write y as a linear combination of the other points listed.
2) v1 = , v2 = , v3 = , y =
A) y = -5v1 + 4v2 + 2v3
B) y = v1 – 2v2 + 2v3
C) y = 5v1 + 3v2 – 7v3
D) y = 4v1 – 2v2 – v3
Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (2.3) Application to Spanning and Linear Independence
Skill: Applied
Objective: Determine if a vector is in the span of a set

3) v1 = , v2 = , v3 = , y =
A) y = -2v1 – 2v2 + 5v3
B) y = 4v1 + 2v2 – 5v3
C) y = 7v1 + 2v2 – 5v3
D) y = 2v1 – 5v2 + 4v3
Answer: D
Diff: 3 Type: BI Var: 21
Topic: (2.3) Application to Spanning and Linear Independence
Skill: Applied
Objective: Determine if a vector is in the span of a set
4) Find all values of h such that y will be in the subspace of R3 spanned by v1, v2, v3
if , , and
A) all h ≠ -14
B) h = -14 or 0
C) h = -14
D) h = -28
Answer: C
Diff: 3 Type: BI Var: 49
Topic: (2.3) Application to Spanning and Linear Independence
Skill: Applied
Objective: Determine if a vector is in the span of a set

Find a homogeneous system which represents a spanning set
1) Find a homogeneous system that defines the set Span .

A) 2×1 – 2×2 + x3 -x4 = 0
B) 2×1 – x2 + x3 – 2×4 = 0
C) x1 – x2 + 2×3 – 2×4 = 0
D) x1 – 2×2 + 2×3 – 2×4 = 0
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (2.3) Application to Spanning and Linear Independence
Skill: Applied
Objective: Find a homogeneous system which represents a spanning set

Determine if a set is a basis for a given hyperplane

1) Find a basis for the plane -3×1 + 2×2 + x3 = 0.
A)

B)

C)

D)
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (2.3) Application to Spanning and Linear Independence
Skill: Applied
Objective: Determine if a set is a basis for a given hyperplane

2) Determine whether the given set is a basis for the given hyperplane.
a) for x1 – x2 + 2×3 – 2×4 = 0.

b) for x1 + 2×3 = 0.

A) Neither a nor b
B) Both a and b
C) a only
D) b only
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (2.3) Application to Spanning and Linear Independence
Skill: Applied
Objective: Determine if a set is a basis for a given hyperplane

Determine if the set is linearly independent.

1) Let Β = and C = .
Then
A) Only C is linearly independent.
B) Both Β and C are linearly independent.
C) Only Β is linearly independent.
D) Both Β and C are linearly dependent.
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (2.3) Application to Spanning and Linear Independence
Skill: Applied
Objective: Determine if a set is linearly independent

Solve the problem.
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: (2.3) Application to Spanning and Linear Independence
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: (2.3) Application to Spanning and Linear Independence
Skill: Applied
Objective: Determine if a set is linearly independent

Determine if a set is a basis for Rn

Determine whether the set of vectors is a basis for R3.
1) 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: MC Var: 1
Topic: (2.3) Application to Spanning and Linear Independence
Skill: Applied
Objective: Determine if a set is a basis for Rn

2) 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: MC Var: 1
Topic: (2.3) Application to Spanning and Linear Independence
Skill: Applied
Objective: Determine if a set is a basis for Rn

Determine whether the given set is a basis for R3.

3) Consider B = and C = .
Then
A) neither of them is a basis for R3.
B) only C is a basis for R3.
C) both B and C are bases for R3.
D) only B is a basis for R3.
Answer: D
Diff: 3 Type: MC Var: 1
Topic: (2.3) Application to Spanning and Linear Independence
Skill: Applied
Objective: Determine if a set is a basis for Rn

Demonstrate understanding of theory.

Solve the problem.
1) Let v1 = , v2 = , v3 = , and H = Span .

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:
B:
C:
D:
A) B, C, and D
B) B only
C) A only
D) B and C
Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (2.3) Application to Spanning and Linear Independence
Skill: Applied
Objective: Demonstrate understanding of theory.

2.4 Applications of Systems of Linear Equations

Resistor Circuits in Electricity

1) Determine the augmented matrix of the system of linear equations, and determine the loop current indicated in the diagram:

A) =

B) =

C) =

D) =
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (2.4) Applications of Systems of Linear Equations
Skill: Applied
Objective: Resistor circuits in electricity

Planar Trusses

1) Determine the system of linear equations for the reaction forces and axial forces in members for the truss shown in the diagram. Assume that all triangles in this diagram are equilateral.

A) = Fv .

B) = Fv .

C) = Fv .

D) = Fv .

Answer: D
Diff: 3 Type: BI Var: 1
Topic: (2.4) Applications of Systems of Linear Equations
Skill: Applied
Objective: Planar trusses

Linear Programming

Find vectors b and c and matrix A so that the problem is set up as a canonical linear programming problem of the form: maximize x subject to Ax ≤ b and x ≥ 0. Do not find the solution.
1) Maximize 2×1 + x2 + 5×3
subject to x1 + 5×2 – 2×3 ≤ 27
-2×2 + 6×3 ≤ 20
and x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
A) b = , c = , and A =
B) b = , c = , and A =
C) b = , c = , and A =
D) b = , c = , and A =
Answer: A
Diff: 2 Type: BI Var: 50+
Topic: (2.4) Applications of Systems of Linear Equations
Skill: Applied
Objective: Linear programming

Solve the linear programming problem.
2) Maximize 6×1 + 7×2
subject to 2×1 + 3×2 ≤ 12
2×1 + x2 ≤ 8
and x1 ≥ 0, x2 ≥ 0
A) maximum = 39 when x1 = 3 and x2 = 3
B) maximum = 28 when x1 = 0 and x2 = 4
C) maximum = 38 when x1 = 4 and x2 = 2
D) maximum = 32 when x1 = 3 and x2 = 2
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (2.3) Application to Spanning and Linear Independence
Skill: Applied
Objective: Linear programming

Solve the word problem by creating and solving a system of linear equations.

3) Alan wants to invest a total of $21,000 in mutual funds and a certificate of deposit (CD). He wants to invest no more in mutual funds than half the amount he invests in the CD. His expected return on mutual funds is 10% and on the CD is 4%. How much money should Alan invest in each area in order to have the largest return on his investments? What is his maximum one-year return?
A) Maximum one-year return is $1,260 when he invests $7,000 in mutual funds and $14,000 in the CD.
B) Maximum one-year return is $840 when he invests $0 in mutual funds and $21,000 in the CD.
C) Maximum one-year return is $2,100 when he invests $21,000 in mutual funds and $0 in the CD.
D) Maximum one-year return is $1,680 when he invests $14,000 in mutual funds and $7,000 in the CD.
Answer: A
Diff: 3 Type: BI Var: 42
Topic: (2.4) Applications of Systems of Linear Equations
Skill: Applied
Objective: Linear programming

4) A store makes two different types of smoothies by blending different fruit juices. Each bottle of Orange Daze smoothie contains 10 fluid ounces of orange juice, 4 fluid ounces of pineapple juice, and 2 fluid ounces of blueberry juice. Each bottle of Pineapple Blue smoothie contains 5 fluid ounces of orange juice, 6 fluid ounces of pineapple juice, and 4 fluid ounces of blueberry juice. The store has 500 fluid ounces of orange juice, 360 fluid ounces of pineapple juice, and 250 fluid ounces of blueberry juice available to put into its smoothies. The store makes a profit of $1.50 on each bottle of Orange Daze and $1 on each bottle of Pineapple Blue. How many bottles of each smoothie should the store make to maximize its profit? What is the maximum profit?
A) Maximum profit is $87.50 when the store makes 35 bottles of Orange Daze and 35 bottles of Pineapple Blue.
B) Maximum profit is $75 when the store makes 50 bottles of Orange Daze and 0 bottles of Pineapple Blue.
C) Maximum profit is $80 when the store makes 40 bottles of Orange Daze and 20 bottles of Pineapple Blue.
D) Maximum profit is $85 when the store makes 30 bottles of Orange Daze and 40 bottles of Pineapple Blue.
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (2.4) Applications of Systems of Linear Equations
Skill: Applied
Objective: Linear programming

5) Daniel decides to feed his cat a combination of two foods: Max Cat and Mighty Cat. He wants his cat to receive four nutritional factors each month. The amounts of these factors (a, b, c, and d) contained in one bag of each food are shown in the chart, together with the total amounts needed.

The costs per bag are $40 for Max Cat and $35 for Mighty Cat. How many bags of each food should be blended to meet the nutritional requirements at the lowest cost? What is the minimum cost?
A) Minimum cost = $610 when he blends 3 bags of Max Cat and 14 bags of Mighty Cat.
B) Minimum cost = $541.25 when he blends bags of Max Cat and bags of Mighty Cat.
C) Minimum cost = $700 when he blends 0 bags of Max Cat and 20 bags of Mighty Cat.
D) Minimum cost = $500 when he blends bags of Max Cat and bags of Mighty Cat.
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (2.4) Applications of Systems of Linear Equations
Skill: Applied
Objective: Linear programming

6) A furniture company makes two different types of lamp stand. Each lamp stand A requires 20 minutes for sanding, 48 minutes for assembly, and 6 minutes for packaging. Each lamp stand B requires 9 minutes for sanding, 32 minutes for assembly, and 8 minutes for packaging. The total number of minutes available each day in each department are as follows: for sanding 3600 minutes, for assembly 9600 minutes, and for packaging 2000 minutes. The profit on each lamp stand A is $30 and the profit on each lamp stand B is $22. How many of each type of lamp stand should the company make per day to maximize their profit? What is the maximum profit?
A) Maximum profit is $5400 when they make 180 of lamp stand A and 0 of lamp stand B.
B) Maximum profit is $6164 when they make 138 of lamp stand A and 92 of lamp stand B.
C) Maximum profit is $6850 when they make 100 of lamp stand A and 175 of lamp stand B.
D) Maximum profit is $6380 when they make 66 of lamp stand A and 200 of lamp stand B.
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (2.4) Applications of Systems of Linear Equations
Skill: Applied
Objective: Linear programming

7) The Acme Class Ring Company designs and sells two types of rings: the VIP and the SST. They can produce up to 24 rings each day using up to 60 total man-hours of labor. It takes 3 man-hours to make one VIP ring and 2 man-hours to make one SST ring. How many of each type of ring should be made daily to maximize the company’s profit, if the profit on a VIP ring is $40 and on an SST ring is $35?
A) 16 VIP and 8 SST
B) 14 VIP and 10 SST
C) 12 VIP and 12 SST
D) 18 VIP and 6 SST
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (2.4) Applications of Systems of Linear Equations
Skill: Applied
Objective: Linear programming

8) An airline with two types of airplanes, P1 and P2, has contracted with a tour group to provide transportation for a minimum of 400 first class, 750 tourist class, and 1500 economy class passengers. For a certain trip, airplane P1 costs $10,000 to operate and can accommodate 20 first class, 50 tourist class, and 110 economy class passengers. Airplane P2 costs $8500 to operate and can accommodate 18 first class, 30 tourist class, and 44 economy class passengers. How many of each type of airplane should be used in order to minimize the operating cost?
A) 9P1 planes and 13P2 planes
B) 9P1 planes and 0P2 planes
C) 11P1 planes and 7P2 planes
D) 7P1 planes and 11P2 planes
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (2.4) Applications of Systems of Linear Equations
Skill: Applied
Objective: Linear programming

9) A summer camp wants to hire counselors and aides to fill its staffing needs at minimum cost. The average monthly salary of a counselor is $2400 and the average monthly salary of an aide is $1100. The camp can accommodate up to 45 staff members and needs at least 30 to run properly. They must have at least 10 aides, and may have up to 3 aides for every 2 counselors. How many counselors and how many aides should the camp hire to minimize cost?
A) 12 counselors and 18 aides
B) 27 counselors and 18 aides
C) 35 counselors and 10 aides
D) 18 counselors and 12 aides
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (2.4) Applications of Systems of Linear Equations
Skill: Applied
Objective: Linear programming