Chapter 7 Orthonormal Bases – 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 7 Orthonormal Bases

7.1 Orthonormal Bases and Orthogonal Matrices

Determine if a set of vectors in Rn is orthogonal and if so normalize to make an orthonormal set

Determine if the given set of vectors in Rn is orthogonal and, if so, normalize to make an orthonormal set.

1)

A)

B) The set is not an orthogonal set of vectors.

C)

D)

Answer: C
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Determine if a set of vectors in Rn is orthogonal and if so normalize to make an orthonormal set

2)

A)

B)

C)

D) The set is not an orthogonal set of vectors.

Answer: B
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Determine if a set of vectors in Rn is orthogonal and if so normalize to make an orthonormal set

3)

A)

B)

C) The set is not an orthogonal set of vectors.

D)

Answer: A
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Determine if a set of vectors in Rn is orthogonal and if so normalize to make an orthonormal set

4)

A) The set is not an orthogonal set of vectors.

B)

C)

D)

Answer: C
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Determine if a set of vectors in Rn is orthogonal and if so normalize to make an orthonormal set
5)

A)

B)

C) The set is not an orthogonal set of vectors.

D)
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Determine if a set of vectors in Rn is orthogonal and if so normalize to make an orthonormal set

6)

A)

B)

C) The set is not an orthogonal set of vectors.

D)

Answer: C
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Determine if a set of vectors in Rn is orthogonal and if so normalize to make an orthonormal set

7)

A) The set is not an orthogonal set of vectors.

B)

C)

D)

Answer: A
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Determine if a set of vectors in Rn is orthogonal and if so normalize to make an orthonormal set

8)

A)

B)

C)

D) The set is not an orthogonal set of vectors.

Answer: C
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Determine if a set of vectors in Rn is orthogonal and if so normalize to make an orthonormal set

9)

A)

B)

C)

D) The set is not an orthogonal set of vectors.

Answer: D
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Determine if a set of vectors in Rn is orthogonal and if so normalize to make an orthonormal set

10)

A) The set is not an orthogonal set of vectors.

B) The set of vectors are orthonormal.

C)

D)

Answer: B
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Determine if a set of vectors in Rn is orthogonal and if so normalize to make an orthonormal set

11)

A) The set is not an orthogonal set of vectors.

B)

C) The set of vectors are orthonormal.

D)

Answer: B
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Determine if a set of vectors in Rn is orthogonal and if so normalize to make an orthonormal set

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

Given an orthonormal basis B and a vector . Find the B-coordinate vector B.
1) B = and =

A)

B)

C)

D)

Answer: D
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Recall
Objective: Find the coordinates of a vector with respect to a orthonormal basis

2) B = and =

A)

B)

C)

D)

Answer: C
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Recall
Objective: Find the coordinates of a vector with respect to a orthonormal basis

3) B = and =

A)

B)

C)

D)

Answer: C
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Recall
Objective: Find the coordinates of a vector with respect to a orthonormal basis

4) B = and =

A)

B) –

C)

D)

Answer: B
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Recall
Objective: Find the coordinates of a vector with respect to a orthonormal basis

5) B = and = .

A)

B)

C)

D)

Answer: D
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Recall
Objective: Find the coordinates of a vector with respect to a orthonormal basis

6) B = and = .

A)

B)

C)

D)

Answer: C
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Recall
Objective: Find the coordinates of a vector with respect to a orthonormal basis

7) Let B = , and , ∈ ℛ4 such that B = and
B = . Determine 2 and ∘ .
A) 30, 18
B) 6, 14
C) 30, 14
D) 6, 18
Answer: C
Diff: 1 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Find the coordinates of a vector with respect to a orthonormal basis

Determine if a matrix is orthogonal

Decide whether the matrix A is orthogonal. If A is not orthogonal, indicate how the columns of A fail to form an orthonormal set.

1) A =
A) A is not orthogonal. The first and third columns are not orthogonal.
B) A is not orthogonal. The second and third columns are not orthogonal.
C) A is orthogonal.
D) A is not orthogonal. The columns are not unit vectors.
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Determine if a matrix is orthogonal

2) A =
A) A is not orthogonal. The columns are not orthogonal and not unit vectors.
B) A is not orthogonal. The columns are not orthogonal.
C) A is orthogonal.
D) A is not orthogonal. The columns are not unit vectors.
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Determine if a matrix is orthogonal

3) A =
A) A is not orthogonal. The columns are not orthogonal and not unit vectors.
B) A is not orthogonal. The columns are not orthogonal.
C) A is orthogonal.
D) A is not orthogonal. The columns are not unit vectors.
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Determine if a matrix is orthogonal
4) A =
A) A is not orthogonal. The columns are not orthogonal and not unit vectors.
B) A is not orthogonal.The columns are not orthogonal.
C) A is orthogonal.
D) A is not orthogonal. The columns are not unit vectors.
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Determine if a matrix is orthogonal

5) A =
A) A is not orthogonal. The columns are not orthogonal and not unit vectors.
B) A is not orthogonal. The columns are not orthogonal.
C) A is orthogonal.
D) A is not orthogonal. The columns are not unit vectors.
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Determine if a matrix is orthogonal

6) A = , where a and b are non-zero real numbers.
A) A is not orthogonal. The columns are not orthogonal and not unit vectors.
B) A is not orthogonal. The columns are not orthogonal.
C) A is orthogonal.
D) A is not orthogonal. The columns are not unit vectors
Answer: D
Diff: 2 Type: BI Var: 1
Topic: (7.1) Orthonormal Bases and Orthogonal Matrices
Skill: Applied
Objective: Determine if a matrix is orthogonal

7.2 Projections and the Gram-Schmidt Procedure

Find the Projection of a vector onto a Subspace

Let W be the subspace spanned by the u’s. Write y as the sum of a vector in W and a vector orthogonal to W.
1) y = , u1 = , u2 =

A) y = +

B) y = +

C) y = +

D) y = +

Answer: A
Diff: 2 Type: BI Var: 16
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Find the projection of a vector onto a subspace

2) y = , u1 = , u2 =

A) y = +

B) y = +

C) y = +

D) y = +

Answer: B
Diff: 2 Type: BI Var: 50+
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Find the projection of a vector onto a subspace

Find the closest point to y in the subspace W spanned by u1 and u2.

3) y = , u1 = , u2 =

A)

B)

C)

D)

Answer: C
Diff: 2 Type: BI Var: 50+
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Find the projection of a vector onto a subspace

4) y = , u1 = , u2 =

A)

B)

C)

D)

Answer: B
Diff: 2 Type: BI Var: 42
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Find the projection of a vector onto a subspace

Determine the projection of onto the subspace spanned by the given orthogonal set.

5) = Β=

A)

B)

C)

D)

Answer: C
Diff: 2 Type: BI Var: 1
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Find the projection of a vector onto a subspace
6) = Β =

A)

B)

C)

D)

Answer: D
Diff: 2 Type: BI Var: 1
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Find the projection of a vector onto a subspace

Find a basis for the orthogonal complement of a subspace

1) Let W = Span . Find the orthogonal complement of W, W⊥ in ℛ4.

A) Span

B) Span

C) Span

D) Span

Answer: D
Diff: 2 Type: BI Var: 1
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Find a basis for the orthogonal complement of a subspace

2) Let W = Span . Find the orthogonal complement of W, W⊥ in ℛ4.

A) Span

B) Span

C) Span

D) Span

Answer: B
Diff: 2 Type: BI Var: 1
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Find a basis for the orthogonal complement of a subspace

3) Let W = Span . Find the orthogonal complement of W, W⊥ in ℛ3.

A) Span

B) Span

C) Span

D) Span

Answer: C
Diff: 2 Type: BI Var: 1
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Find a basis for the orthogonal complement of a subspace

4) Let W = Span . Find the orthogonal complement of W, W⊥ in ℛ3.

A) Span

B) Span

C) Span

D) Span

Answer: A
Diff: 2 Type: BI Var: 1
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Find a basis for the orthogonal complement of a subspace

5) Let W = Span . Find the orthogonal complement of W, W⊥ in ℛ2.

A) Span

B) Span

C) Span

D) Span

Answer: D
Diff: 1 Type: BI Var: 1
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Find a basis for the orthogonal complement of a subspace

6) Let W = Span . Find the orthogonal complement of W, W⊥ in ℛ4.

A) Span

B) Span

C) Span

D) Span

Answer: D
Diff: 2 Type: BI Var: 1
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Find a basis for the orthogonal complement of a subspace

Use the Gram-Schimdt procedure to produce an orthogonal basis from a spanning set

The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.

1) Let x1 = , x2 =

A) ,

B) ,

C) ,

D) ,

Answer: A
Diff: 3 Type: BI Var: 50+
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Use the Gram-Schimdt procedure to produce an orthogonal basis from a spanning set

2) Let x1 = , x2 = , x3 =
A) , ,

B) , ,

C) , ,

D) , ,

Answer: B
Diff: 3 Type: BI Var: 1
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Use the Gram-Schimdt procedure to produce an orthogonal basis from a spanning set

Find a QR factorization of the matrix A.

3) A =

A) Q = , R =

B) Q = , R =

C) Q = , R =

D) Q = , R =

Answer: A
Diff: 3 Type: BI Var: 48
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Use the Gram-Schimdt procedure to produce an orthogonal basis from a spanning set

4) A =

A) Q = , R =

B) Q = , R =

C) Q = , R =

D) Q = , R =

Answer: C
Diff: 3 Type: BI Var: 1
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Use the Gram-Schimdt procedure to produce an orthogonal basis from a spanning set

The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthonormal basis for W.

5)

A)

B)

C)

D)

Answer: C
Diff: 3 Type: BI Var: 1
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Use the Gram-Schimdt procedure to produce an orthogonal basis from a spanning set

6)

A)

B)

C)

D)

Answer: D
Diff: 3 Type: BI Var: 1
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Use the Gram-Schimdt procedure to produce an orthogonal basis from a spanning set

7)

A)

B)

C)

D)

Answer: B
Diff: 3 Type: BI Var: 1
Topic: (7.2) Projections and the Gram-Schmidt Procedure
Skill: Applied
Objective: Use the Gram-Schimdt procedure to produce an orthogonal basis from a spanning set

7.3 Method of Least Squares

Find a line or quadratic of best fit

Find the equation y = + x of the least-squares line that best fits the given data points.

1) Data points: (5, -3), (2, 2), (4, 3), (5, -1)

x = , y =
A) y = + 0x
B) y = – x
C) y = – x
D) y = – 2x
Answer: C
Diff: 2 Type: BI Var: 16
Topic: (7.3) Method of Least Squares
Skill: Applied
Objective: Find a line or quadratic of best fit
2) Data points: (2, 1), (3, 2), (7, 3), (8, 4)

x = , y =
A) y = – + x
B) y = + x
C) y = + x
D) y = – 3 + x
Answer: B
Diff: 2 Type: BI Var: 5
Topic: (7.3) Method of Least Squares
Skill: Applied
Objective: Find a line or quadratic of best fit

Find the equation y = at2 + bt of the least-squares line that best fits the given data points.

3) Data points: (-1,4), (0,1), (1,1)
A) y = – t2 + t
B) y = t2 – t
C) y = – t2 – t
D) y = t2 + t
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (7.3) Method of Least Squares
Skill: Applied
Objective: Find a line or quadratic of best fit
Find the least-squares line y = + x that best fits the given data.

4) Given: The data points (-3, 2), (-2, 5), (0, 5), (2, 2), (3, 7).
Suppose the errors in measuring the y-values of the last two data points are greater than for the other points. Weight these data points half as much as the rest of the data.

X = , β = , y =
A) y = 4.6 + 0.45x
B) y = 0.9 + 1.54x
C) y = 4.9 + 1.54x
D) y = 0.45 + 4.6x
Answer: A
Diff: 2 Type: BI Var: 24
Topic: (7.3) Method of Least Squares
Skill: Applied
Objective: Find a line or quadratic of best fit

5) Given: The data points (-2, 2), (-1, 5), (0, 5), (1, 2), (2, 2).
Suppose the errors in measuring the y-values of the last two data points are greater than for the other points. Weight these data points twice as much as the rest of the data.

X = , β = , y =
A) y = 2.6 – 0.53x
B) y = 5.8 – 0.90x
C) y = 2.9 – 0.45x
D) y = 2.8 – 0.55x
Answer: C
Diff: 2 Type: BI Var: 24
Topic: (7.3) Method of Least Squares
Skill: Applied
Objective: Find a line or quadratic of best fit
Find a best approximation solution to an inconisistent overdetermined system

Given A and b, determine the least-squares error in the least-squares solution of

1) A = , b =
A) 1.63299316
B) 3.82970843
C) 91.929925
D) 260.495468
Answer: A
Diff: 3 Type: BI Var: 12
Topic: (7.3) Method of Least Squares
Skill: Applied
Objective: Find a best approximation solution to an inconsistent overdetermined system

Find a least-squares solution of the inconsistent system A = , that minimizes .

2) A = , b =
A)

B)

C)

D)

Answer: A
Diff: 3 Type: BI Var: 32
Topic: (7.3) Method of Least Squares
Skill: Applied
Objective: Find a best approximation solution to an inconsistent overdetermined system

3) A = , b =
A) + x4

B) + x4

C) + x4

D) + x4

Answer: B
Diff: 3 Type: BI Var: 1
Topic: (7.3) Method of Least Squares
Skill: Applied
Objective: Find a best approximation solution to an inconsistent overdetermined system

7.4 Inner Product Spaces

Evaluate an inner product in various inner product spaces

For p(t), q(t) ∈ P2, define = p(t0) q(t0) + p(t1) q(t1) + p(t2)q(t2), where t0, t1, t2 are distinct real numbers. Compute the length of the given vector.
1) p(t) = 10t2 and q(t) = t – 1, where t0 = 0, t1 = , t2 = 1
A) = ; =
B) = 10 ; =
C) = 0.5 ; =
D) = ; = 10
Answer: A
Diff: 2 Type: BI Var: 5
Topic: (7.4) Inner Product Spaces
Skill: Recall
Objective: Evaluate an inner product in various inner product spaces

2) p(t) = 4t + 7 and q(t) = 7t – 5, where t0 = 0, t1 = 1, t2 = 2
A) = ; =
B) = ; =
C) = ; =
D) = ; =
Answer: D
Diff: 2 Type: BI Var: 17
Topic: (7.4) Inner Product Spaces
Skill: Applied
Objective: Evaluate an inner product in various inner product spaces

3) For p(t), q(t) ∈ P2, define = p(0) q(0) + p(1) q( 1) + p(2)q(2). Calculate .
A) 4
B)
C) 3
D) 2
Answer: C
Diff: 2 Type: BI Var: 1
Topic: (7.4) Inner Product Spaces
Skill: Applied
Objective: Evaluate an inner product in various inner product spaces

4) Find the norm of A= in M(2,2) under the inner product = tr(BTA).
A) 4
B)
C) 3
D) 2
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (7.4) Inner Product Spaces
Skill: Applied
Objective: Evaluate an inner product in various inner product spaces

Determine if an operator is an inner product on a given vector space

Solve the problem.

1) With the given positive numbers, show that vectors u = (u1, u2)and v = (v1, v2) define an inner product on R2 using the 4 axioms.
Set = 2 u1v1 + 2 u2v2.
Answer:
Axiom 1: = 2 u1v1 + 2 u2v2 = 2 v1u1 + 2 v2u2 =
Axiom 2: If w = (w1,w2), then = 2(u1+ v1) w1 + 2(u2+ v2)w2
= 2u1w1 + 2u2w2 + 2v1w1 + 2v2w2
= +
Axiom 3: = 2c(u1)v1 + 2(cu2)v2 = c(2u1v1 + 2u2v2) = c
Axiom 4: = 2u12 + 2u22 ≥ 0, and 2u12 + 2u22 = 0 only
if u1 = u2 = 0. Also, = 0.
Diff: 2 Type: SA Var: 24
Topic: (7.4) Inner Product Spaces
Skill: Recall
Objective: Determine if an operator is an inner product on a given vector space

2) Let t0, …., tn be distinct real numbers. For p and q in Pn, define t0)q(t0) +P(t1)q(t1) +… + P(tn)q(tn) Show that defines an inner product on Pn.
Answer: Axioms 1-3 are readily checked. For Axiom 4, note that
= [P(t0)]2 + [P(t1)]2 +… + [P(tn)]2≥ 0
Also,
If = 0, then p must vanish at n + 1 points: t0, …., tn. This is possible only if p is the zero polynomial, because the degree of p is less than n + 1.
Diff: 2 Type: SA Var: 1
Topic: (7.4) Inner Product Spaces
Skill: Recall
Objective: Determine if an operator is an inner product on a given vector space

3) For f, g in C[a, b], set =
Show that defines an inner product on C[a, b].
Answer: Answers will vary.
Inner product Axioms 1-3 follow from elementary properties of definite integrals.
Axiom 4: = dt ≥ 0.
The function [f(t)]2 is continuous and nonnegative on [a, b]. If the definite integral of [f(t)]2 is zero, then [f(t)]2 must be identically zero on [a, b]. Thus, = 0 implies that f is the zero function on
Diff: 2 Type: SA Var: 1
Topic: (7.4) Inner Product Spaces
Skill: Recall
Objective: Determine if an operator is an inner product on a given vector space

4) Determine which of the following defines an inner product on P2.
I) For p(t), q(t) ∈ P2, define = p(0) q(0) + p(1) q( 1).
II) For p(t), q(t) ∈ P2, define = p(0) q(0) + p(1) q( 1) + p(2)q(2).
A) Both I and II
B) II only
C) I only
D) Neither I nor II
Answer: B
Diff: 2 Type: BI Var: 1
Topic: (7.4) Inner Product Spaces
Skill: Recall
Objective: Determine if an operator is an inner product on a given vector space
Use the Gram-Schimdt procedure to produce an orthogonal basis from a spanning set in an inner product Space

Solve the problem.
1) Let V be in P4 with inner product defined as <p,q> = p(t0)q(t0) + p(t1)q(t1) + p(t2)q(t2) + p(t3)q(t3) + p(t4)q(t4), involving evaluation of polynomials at -3, -1, 0, 1, and 3, and view P2 as subspace of V. Produce an orthogonal basis for P2 by applying the Gram-Schmidt process to the polynomials 1, t, and t2.
A) 1, t, t2 – 4
B) 1, t, t2 + 4
C) 1, t, t2 +
D) 1, t, t2 –
Answer: A
Diff: 3 Type: BI Var: 3
Topic: (7.4) Inner Product Spaces
Skill: Applied
Objective: Use the Gram-Schimdt procedure to produce an orthogonal basis from a spanning set in an inner product space.

2) Let V be in P4with inner product defined as
<p,q> = p(t0)q(t0) + p(t1)q(t1) + p(t2)q(t2) + p(t3)q(t3) + p(t4)q(t4), involving evaluation of polynomials
at -6, -3, 0, 3, and 6, and view P2 as subspace of V. Produce an orthogonal basis for P2 by applying the Gram-Schmidt process to the polynomials 1, t, and t2.
A) 1, t, t2 +18
B) 1, t, t2 -18
C) 1, t, t2 –
D) 1, t, t2 –
Answer: B
Diff: 3 Type: BI Var: 9
Topic: (7.4) Inner Product Spaces
Skill: Applied
Objective: Use the Gram-Schimdt procedure to produce an orthogonal basis from a spanning set in an inner product space.
3) Define the inner product = 2x1y1 + x2y2 + 3x3y3 on ℛ3. Use the Gram-Schimdt procedure to produce an orthogonal basis for Σ = .
A)

B)

C)

D)

Answer: C
Diff: 3 Type: BI Var: 1
Topic: (7.4) Inner Product Spaces
Skill: Applied
Objective: Use the Gram-Schimdt procedure to produce an orthogonal basis from a spanning set in an inner product space.

Solve the problem.
4) Let V be the space C[0, 1] and let W be the subspace spanned by the polynomials and Use the Gram-Schmidt process to find an orthogonal basis for W.
Answer: As a function, (t) = 12t2 – 48t + 20. The orthogonal basis for the subspace W is .
Diff: 3 Type: SA Var: 4
Topic: (7.4) Inner Product Spaces
Skill: Applied
Objective: Use the Gram-Schimdt procedure to produce an orthogonal basis from a spanning set

7.5 Fourier Series

Solve the problem.
1) Let C[0, π] have the inner product = and let m and n be unequal positive integers. Prove that cos(mt) and cos(nt) are orthogonal.
A) =
=
= from [0, π]
= 0.
B) =
=
= from [0, π]
= 1.
C) =
=
= from [0, π]
= 0.
D) =
=
= from [0, π]
= 1.
Answer: A
Diff: 3 Type: BI Var: 1
Topic: (7.5) Fourier Series
Skill: Applied
Objective: Fourier Series

2) Find the nth-order Fourier approximation to the function f(t) = 4t on the interval [0, 2π].
A) 4π – 8sin(t) – 4sin(2t) – sin(3t) – … – sin(nt)
B) 4π – 8sin(t) – 4sin(2t) – sin(3t) – … – sin(nt)
C) 4π – 8cos(t) – 4sin(2t) – cos(3t) – … – cos(nt)
D) π – cos(t) – cos(2t) – cos(3t) – … – cos(nt)
Answer: A
Diff: 3 Type: BI Var: 4
Topic: (7.5) Fourier Series
Skill: Applied
Objective: Fourier Series