Chapter 9 Complex 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 9 Complex Vector Spaces

9.1 Complex Numbers

Identify the real and imaginary parts of a complex number and take complex conjugates

1) Determine the complex conjugate of 7i – 3.
A) -3 – 7i
B) 3 + 7i
C) -3 + 7i
D) 3 – 7i
Answer: A
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Recall
Objective: Identify the real and imaginary parts of a complex number and take complex conjugates

2) Determine the complex conjugate of 7i + 5.
A) -5 – 7i
B) 5 + 7i
C) -5 + 7i
D) 5 – 7i
Answer: D
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Recall
Objective: Identify the real and imaginary parts of a complex number and take complex conjugates

Determine the real and imaginary parts of the complex number.

3) z = (5 – 3i) (5 + 3i)
A) Re(z) = 30 and Im(z) = 4
B) Re(z) = 32 and Im(z) = 2
C) Re(z) = 34 and Im(z) = 6
D) Re(z) = 34 and Im(z) = 0
Answer: D
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Recall
Objective: Identify the real and imaginary parts of a complex number and take complex conjugates

4) z = (3 – 2i) (-2 + 5i)
A) Re(z) = -16 and Im(z) = 11
B) Re(z) = 4 and Im(z) = 19
C) Re(z) = 16 and Im(z) = 19
D) Re(z) = 4 and Im(z) = 11
Answer: B
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Recall
Objective: Identify the real and imaginary parts of a complex number and take complex conjugates
5)
A) Re(z) = 0 and Im(z) = 1
B) Re(z) = 1 and Im(z) = 0
C) Re(z) = 0 and Im(z) = -1
D) Re(z) = -1 and Im(z) = 0
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Recall
Objective: Identify the real and imaginary parts of a complex number and take complex conjugates

Perform simple operations on complex numbers

1) Calculate (3 – 4i) – (2 – 5i).
A) 1 + i
B) 1 – 9i
C) 5 + 9i
D) 5 + i
Answer: A
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Perform simple operations on complex numbers

2) Calculate i6.
A) 1
B) -1
C) i
D) -i
Answer: B
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Perform simple operations on complex numbers

3) Calculate (5 – 3i) (5 + 3i).
A) 8
B) 16
C) 34
D) 2
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Perform simple operations on complex numbers
4) Calculate (6 + 2i)i.
A) -2 – 6i
B) 2 + 6i
C) 2 – 6i
D) -2 + 6i
Answer: D
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Perform simple operations on complex numbers

5) Calculate (3 – 4i) (2 – 3i) – (2 – 3i).
A) -8 – 14i
B) -16 – 14i
C) 3 – 4i
D) 0
Answer: A
Diff: 2 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Perform simple operations on complex numbers

6) Calculate (7 – 2i) (-3 + 3i).
A) -15 – 27i
B) -27 + 27i
C) -15 + 27i
D) -21 – 6i
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Perform simple operations on complex numbers

7) Express in the standard form.
A) – i
B) 1
C) 1 + i
D) + i
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Perform simple operations on complex numbers
8) Express in the standard form.
A) – i
B) – i
C) + i
D) + i
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Perform simple operations on complex numbers

Use polar form to determine products and quotients of complex numbers

Use polar form to calculate the following:
1) (1 + i)3
A) -2 – 2i
B) 2 – 2i
C) -2 + 2i
D) 2 + 2i
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Use polar form to determine products and quotients of complex numbers

2) (1 + i) (1 + i)

A)
B) 2
C)
D) 2
Answer: B
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Use polar form to determine products and quotients of complex numbers
3)
A)
B)
C)
D)
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Use polar form to determine products and quotients of complex numbers

4)
A) 2
B) 2
C) 2
D) 2
Answer: B
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Use polar form to determine products and quotients of complex numbers

5) 9
A) -64
B) 512
C) -512
D) 64
Answer: B
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Use polar form to determine products and quotients of complex numbers
6) 8
A) -128
B) 128
C) 128
D) 128
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Use polar form to determine products and quotients of complex numbers

7) (1 + i)4
A) 4
B) -4i
C) -4
D) 4i
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Use polar form to determine products and quotients of complex numbers

8) (3 – 3i)3
A) 54 + 54i
B) -54 – 54i
C) -54 + 54i
D) 54 – 54i
Answer: B
Diff: 1 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Use polar form to determine products and quotients of complex numbers

Use polar form to find the roots of a complex number

1) (-1)1/6
A) , 0 ≤ k ≤ 5
B) , 0 ≤ k ≤ 5
C) , 0 ≤ k ≤ 5
D) 2 , 0 ≤ k ≤ 5
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Use polar form to find the roots of a complex number
2) (-i)1/4
A) , 0 ≤ k ≤ 3
B) , 0 ≤ k ≤ 3
C) , 0 ≤ k ≤ 4
D) , 0 ≤ k ≤ 3
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Use polar form to find the roots of a complex number

3) (1)1/2
A) ± 1
B) ± 1, i
C) ± i
D) ±1, ±i
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Use polar form to find the roots of a complex number

4) (-1)1/3
A) 1, -1, and i
B) 1, , and
C) 1, , and
D) 1, , and
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Use polar form to find the roots of a complex number
5) 1/2
A) and –
B) and –
C) and –
D) and –
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (9.1) Complex Numbers
Skill: Applied
Objective: Use polar form to find the roots of a complex number

9.2 Systems with Complex Numbers

Solve a system of linear equations with complex coefficients

1) ix + y = 3
ix – y = 2
A) =
B) =
C) =
D) =
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (9.2) Systems with Complex Numbers
Skill: Applied
Objective: Solve a system of linear equations with complex coefficients
2) -ix + y = 0
x + iy = 0

A) = t, t ∈ C
B) = t, t ∈ C
C) = t, t ∈ C
D) = t, t ∈ C
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (9.2) Systems with Complex Numbers
Skill: Applied
Objective: Solve a system of linear equations with complex coefficients

3) -ix + y = 2
x + iy = 1

A) = t, t ∈ C
B) = t, t ∈ C
C) No solution
D) = t, t ∈ C
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (9.2) Systems with Complex Numbers
Skill: Applied
Objective: Solve a system of linear equations with complex coefficients

4) (1 – i) x + y = 0
2x +(1 + i) y = 0

A) = t, t ∈ C
B) = t, t ∈ C
C) = t, t ∈ C
D) = t, t ∈ C
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (9.2) Systems with Complex Numbers
Skill: Applied
Objective: Solve a system of linear equations with complex coefficients

5) (1 + i) z1 + 2 i z2 = 1
(1 + i) z2 + z3 = – i
z1 – z3 = 0

A) = + t
B) = + t
C) = + t
D) = + t
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (9.2) Systems with Complex Numbers
Skill: Applied
Objective: Solve a system of linear equations with complex coefficients

9.3 Vector Spaces over C

Perform linear combinations of vectors in Cn

Calculate the following:
1) –

A)
B)
C)
D)
Answer: D
Diff: 1 Type: BI Var: 1
Topic: (9.3) Vector Spaces over C
Skill: Recall
Objective: Perform linear combinations of vectors in Cn

2) +

A)
B)
C)
D)
Answer: B
Diff: 1 Type: BI Var: 1
Topic: (9.3) Vector Spaces over C
Skill: Recall
Objective: Perform linear combinations of vectors in Cn
3) -3i

A)
B)
C)
D)
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (9.3) Vector Spaces over C
Skill: Recall
Objective: Perform linear combinations of vectors in Cn

4) -2i

A)
B)
C)
D)
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (9.3) Vector Spaces over C
Skill: Recall
Objective: Perform linear combinations of vectors in Cn

Find a basis for special subspaces in Cn

1) Find a basis for the column space of A = .

A)
B)
C)
D)
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (9.3) Vector Spaces over C
Skill: Applied
Objective: Find a basis for special subspaces in Cn

2) Find a basis for the row space of A = .

A)
B)
C)
D)
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (9.3) Vector Spaces over C
Skill: Applied
Objective: Find a basis for special subspaces in Cn

3) Find a basis for the null space of A = .

A)

B)

C) ,
D)
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (9.3) Vector Spaces over C
Skill: Applied
Objective: Find a basis for special subspaces in Cn

4) Find a basis for the column space of A = .
A)
B)
C)
D)

Answer: D
Diff: 2 Type: BI Var: 1
Topic: (9.3) Vector Spaces over C
Skill: Applied
Objective: Find a basis for special subspaces in Cn

5) Find a basis for the row space of A = .

A)
B)
C)
D)
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (9.3) Vector Spaces over C
Skill: Applied
Objective: Find a basis for special subspaces in Cn

6) Find a basis for the null space of A = .

A)
B)

C)
D)
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (9.3) Vector Spaces over C
Skill: Applied
Objective: Find a basis for special subspaces in Cn

9.4 Eigenvectors in Complex Vector Spaces

Find the real canonical form a matrix and the corresponding change of coordinates matrix

Find the eigenvalues of A, and find a basis for each eigenspace.
1) A =

A) 0.6 – 0.8i, ; 0.6 + 0.8i,
B) 0.6 + 0.8i, ; 0.6 – 0.8i,
C) -0.6 + 0.8i, ; -0.6 – 0.8i,
D) -0.6 – 0.8i, ; -0.6 + 0.8i,
Answer: A
Diff: 2 Type: BI Var: 22
Topic: (9.4) Eigenvectors in Complex Vector Spaces
Skill: Applied
Objective: Find the real canonical form a matrix and the corresponding change of coordinates matrix

2) A =
A) 2 – 4i, ; 2 + 4i,
B) 2 – 4i, ; 2 + 4i,
C) 2 – 4i, ; 2 + 4i,
D) 2 – 4i, ; 2 + 4i,
Answer: D
Diff: 3 Type: BI Var: 50+
Topic: (9.4) Eigenvectors in Complex Vector Spaces
Skill: Applied
Objective: Find the real canonical form a matrix and the corresponding change of coordinates matrix

3) Find a real canonical form of a matrix C of A = and find a change of coordinates matrix P such that P-1AP = C.
A) C = and P =
B) C = and P =
C) C = and P =
D) C = and P =
Answer: C
Diff: 3 Type: BI Var: 1
Topic: (9.4) Eigenvectors in Complex Vector Spaces
Skill: Applied
Objective: Find the real canonical form a matrix and the corresponding change of coordinates matrix
4) Find a real canonical form of a matrix C of A = and find a change of coordinates matrix P such that P-1AP = C.

A) P = , and C =
B) P = , and C =
C) P = , and C =
D) P = , and C =
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (9.4) Eigenvectors in Complex Vector Spaces
Skill: Applied
Objective: Find the real canonical form a matrix and the corresponding change of coordinates matrix

5) Find a real canonical form of a matrix C of A = and find a change of coordinates matrix P such that P-1AP = C.

A) P = , and C =
B) P = , and C =
C) P = , and C =
D) P = , and C =
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (9.4) Eigenvectors in Complex Vector Spaces
Skill: Applied
Objective: Find the real canonical form a matrix and the corresponding change of coordinates matrix
9.5 Inner Products in Complex Vector Spaces

Evaluate complex inner products on Cn

Use the standard inner product in Cn to calculate the following:
1) Let = , = . Then find .
A) 2i
B) 2
C) 2 – 2i
D) 0
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (9.5) Inner Products in Complex Vector Spaces
Skill: Recall
Objective: Evaluate complex inner products on Cn

2) Let = . Then find .

A) 4
B) 1
C) 0
D) 2
Answer: D
Diff: 1 Type: BI Var: 1
Topic: (9.5) Inner Products in Complex Vector Spaces
Skill: Recall
Objective: Evaluate complex inner products on Cn

3) Let = . Then find .
A) 1
B)
C)
D) 2
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (9.5) Inner Products in Complex Vector Spaces
Skill: Recall
Objective: Evaluate complex inner products on Cn

4) Let = , and = . Then find proj .

A)
B)
C)
D)
Answer: D
Diff: 1 Type: BI Var: 1
Topic: (9.5) Inner Products in Complex Vector Spaces
Skill: Recall
Objective: Evaluate complex inner products on Cn

5) Let = , and = . Then find all complex values of k (if any) such that and are orthogonal in C3.
A)
B)
C) 2i
D) -2i
Answer: D
Diff: 1 Type: BI Var: 1
Topic: (9.5) Inner Products in Complex Vector Spaces
Skill: Recall
Objective: Evaluate complex inner products on Cn

Determine if a matrix is unitary

1) Which of the following matrices are unitary?
A = and B =

A) Neither A nor B
B) B only
C) A only
D) Both A and B
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (9.5) Inner Products in Complex Vector Spaces
Skill: Recall
Objective: Determine if a matrix is unitary

2) Which of the following statements are true for a unitary matrix A?
I) Det A = ±1.
II) All of the eigenvalues satisfy = 1.
A) Neither I nor II
B) Both I and II
C) II only
D) I only
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (9.5) Inner Products in Complex Vector Spaces
Skill: Applied
Objective: Determine if a matrix is unitary

Use the Gram-Schmidt procedure and find projections using complex inner products

Use the Gram-Schmidt procedure to find an orthogonal basis for S.

1) S = Span .
A)

B)
C)
D)
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (9.5) Inner Products in Complex Vector Spaces
Skill: Applied
Objective: Use the Gram-Schmidt procedure and find projections using complex inner products

2) Determine the projection of = onto the subspace C3 spanned by the orthogonal set of vectors .

A)
B)
C)
D)
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (9.5) Inner Products in Complex Vector Spaces
Skill: Applied
Objective: Use the Gram-Schmidt procedure and find projections using complex inner products

3) Determine the projection of = onto the subspace C3 spanned by the orthogonal set of vectors .

A)
B)
C)
D)
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (9.5) Inner Products in Complex Vector Spaces
Skill: Applied
Objective: Use the Gram-Schmidt procedure and find projections using complex inner products

9.6 Hermitian Matrices and Unitary Diagonalization

Determine if a matrix is Hermitian

1) Which of the following matrices are Hermitian?
A = , B =

A) Both A and B
B) B only
C) A only
D) Neither A nor B
Answer: D
Diff: 1 Type: BI Var: 1
Topic: (9.6) Hermitian Matrices and Unitary Diagonalization
Skill: Recall
Objective: Determine if a matrix is Hermitian

2) Find value(s) of k (k should be a real number) such that A is Hermitian., where A = .
A) k = -1
B) k = 2
C) k = 0
D) k = 1
Answer: A
Diff: 1 Type: BI Var: 1
Topic: (9.6) Hermitian Matrices and Unitary Diagonalization
Skill: Applied
Objective: Determine if a matrix is Hermitian

Unitarily diagonalize the given Hermitian matrix A.

Find a unitary matrix U and a diagonal matrix D such that U∗AU = D.

1) A =

A) U = , D =
B) U = , D =
C) U = , D =
D) U = , D =
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (9.6) Hermitian Matrices and Unitary Diagonalization
Skill: Applied
Objective: Unitarily diagonalize a Hermitian matrix

2) A =

A) U = , D =
B) U = , D =
C) U = , D =
D) U = , D =
Answer: D
Diff: 3 Type: BI Var: 1
Topic: (9.6) Hermitian Matrices and Unitary Diagonalization
Skill: Applied
Objective: Unitarily diagonalize a Hermitian matrix

3) A =

A) U = ,
D =

B) U = ,
D =

C) U = ,
D =

D) U = ,
D =
Answer: B
Diff: 3 Type: BI Var: 1
Topic: (9.6) Hermitian Matrices and Unitary Diagonalization
Skill: Applied
Objective: Unitarily diagonalize a Hermitian matrix