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Poole - Linear Algebra: A Modern Intro 4/e (Homework)

James Finch

Math - College, section 1, Fall 2019

Instructor: Dr. Friendly

Current Score : 16 / 21

Due : Sunday, January 27, 2030 00:00 EST

Last Saved : n/a Saving...  ()

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Question
Points
1 2 3 4 5 6 7 8 9 10
1/1 2/2 2/2 4/4 1/2 1/1 1/2 2/2 0/1 2/4
Total
16/21 (76.2%)

  • Instructions

    David Poole's innovative Linear Algebra: A Modern Introduction, 4th edition, published by Cengage Learning, emphasizes a vectors approach and better prepares students to make the transition from computational to theoretical mathematics. Balancing theory and applications, the book is written in a conversational style and combines a traditional presentation with a focus on student-centered learning. Theoretical, computational, and applied topics are presented in a flexible yet integrated way. Stressing geometric understanding before computational techniques, vectors and vector geometry are introduced early to help students visualize concepts and develop mathematical maturity for abstract thinking. Additionally, the book includes ample applications drawn from a variety of disciplines, which reinforce the fact that linear algebra is a valuable tool for modeling real-life problems.

    Included in this book are unique expandable matrix questions, that allow students to add or remove columns and rows from matrices to better assess their knowledge.

    Questions 1 and 2 are examples of expandable matrix questions.

    Question 1 also has a Watch It, a video showing the solution of a similar problem.

    Question 7 requires the beginning of a proof by contradiction to be provided. This checks if students understand how to write the negation of a statement --- which parts of the original statement are negated and which are kept the same, and what language to use (e.g. "there exists" vs. "any").

    Question 8 provides two proofs with incorrect statements, and the student must determine what mistakes the proof makes. They must determine how many times the proof errs, and what correct reasoning should be substituted.

    Question 9 has a Master It tutorial that walks the student through how to do the problem.

    Question 10 is an Expanded Problem that requires the student to show intermediate answers required to complete the question.

    Click here for a list of all of the questions coded in WebAssign. This demo assignment allows many submissions and allows you to try another version of the same question for practice wherever the problem has randomized values.

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1. 1/1 points  |  Previous Answers PooleLinAlg4 3.1.005. My Notes
Question Part
Points
Submissions Used
1
1/1
5/50
Total
1/1
 
Let
A = 
30
−15
,    B = 
43−4
−50−1
.
Compute the indicated matrix. (If this is not possible, enter DNE in any single blank).
AB

Correct: Your answer is correct.

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2. 2/2 points  |  Previous Answers PooleLinAlg4 3.5.021. My Notes
Question Part
Points
Submissions Used
1 2
1/1 1/1
1/50 1/50
Total
2/2
 
Find bases for row(A) and col(A) in the given matrix using AT.
A = 
10−1
113
row(A)    

Correct: Your answer is correct.
col(A)    

Correct: Your answer is correct.

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3. 2/2 points  |  Previous Answers PooleLinAlg4 3.5.035. My Notes
Question Part
Points
Submissions Used
1 2
1/1 1/1
1/50 1/50
Total
2/2
 
Give the rank and the nullity of the matrix.
A = 
50−5
444
rank(A)  =  Correct: Your answer is correct.
nullity(A)  =  Correct: Your answer is correct.
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4. 4/4 points  |  Previous Answers PooleLinAlg4 3.6.021.MI. My Notes
Question Part
Points
Submissions Used
1 2 3 4
1/1 1/1 1/1 1/1
3/50 3/50 3/50 3/50
Total
4/4
 
Find the standard matrix of the given linear transformation from the set of real numbers2 to the set of real numbers2. (Use only positive angles in your calculations.)
Clockwise rotation through 150° about the origin
cos((360−150)360​·2π)
Correct: Your answer is correct.
−sin((360−150)360​·2π)
Correct: Your answer is correct.
(sin((360−150)360​·2π))
Correct: Your answer is correct.
cos((360−150)360​·2π)
Correct: Your answer is correct.
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5. 1/2 points  |  Previous Answers PooleLinAlg4 3.7.019. My Notes
Question Part
Points
Submissions Used
1 2
1/1 –/1
2/50 0/50
Total
1/2
 
Determine whether the matrix is an exchange matrix.
1/21/8
1/27/8
     Correct: Your answer is correct.

If it is an exchange matrix, find a nonnegative price vector x that satisfies the equation Ex = x. (If it is not an exchange matrix, enter NONE in any single cell.)
x  =
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6. 1/1 points  |  Previous Answers PooleLinAlg4 3.7.053. My Notes
Question Part
Points
Submissions Used
1
1/1
1/50
Total
1/1
 
Determine the adjacency matrix A of the given digraph.
A digraph with 4 vertex is given. The vertex v1 has directed edges that connect it to the vertices v2 and v3. The vertex v2 does not have any directed edges. The vertex v3 has directed edges that connect it to vertices v2 and v4. The vertex v4 has a directed edge that connects it to the vertex v1.
A =
Correct: Your answer is correct.
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7. 1/2 points  |  Previous Answers My Notes
Question Part
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1 2
1/1 –/1
1/50 0/50
Total
1/2
 
Prove that if B is an
n ✕ n
 matrix and E is an
n ✕ n
 elementary matrix, then det(E B) = (det E)(det B).
Which of the following could start a proof by contradiction of this lemma?
     Correct: Your answer is correct.
In your group, discuss the steps necessary to complete the proof. State your steps and reasoning below.

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8. 2/2 points  |  Previous Answers My Notes
Question Part
Points
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1 2
1/1 1/1
2/50 3/50
Total
2/2
 
Prove that if A and B are
n ✕ n
 matrices with A ~ B, then Am ~ Bm for all integers m ≥ 0.
The following proposed proofs may or may not be true. Below the proof, indicate if it is correct or not, and if not, identify the mistake(s) made.
Proposed Proof #1: Suppose that A ~ B, so that A ~ P−1BP. If m > 0, then
\[\textit{A}^{m} = \underbrace{\textit{P}^{-1} \textit{P}^{-1} ... \textit{P}^{-1} \textit{P}^{-1}}_\text{\textit{m} times} \cdot \underbrace{\textit{B} \textit{B} ... \textit{B} \textit{B}}_\text{\textit{m} times} \cdot \underbrace{\textit{P} \textit{P} ... \textit{P} \textit{P}}_\text{\textit{m} times} = \textit{P}^{-m}\textit{B}^{m}\textit{P}^{m}\]
But Am ~ P−mBmPm means that Am ~ Bm.
Is the proof correct? If not, which mistakes were made? (Select all that apply.)
Correct: Your answer is correct.

Proposed Proof #2: Suppose that A ~ B, so that A ~ P−1BP. Then if m = 0, we have Am = A and Bm = B, so that A0 ~ B0. If m > 0, then
\[\textit{A}^{m} = \underbrace{\textit{P}^{-1}\textit{BP} \cdot \textit{P}^{-1}\textit{BP} ... \textit{P}^{-1}\textit{BP} \cdot \textit{P}^{-1}\textit{BP}}_\text{\textit{m} times} = \textit{P}^{-1}\textit{B}\textit{P}\]
But Am ~ P−1BP, and as stated earlier Bm = B. Therefore, Am ~ Bm.
Is the proof correct? If not, which mistakes were made? (Select all that apply.)
Correct: Your answer is correct.

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9. 0/1 points  |  Previous Answers PooleLinAlg4 3.3.057.MI. My Notes
Question Part
Points
Submissions Used
1
0/1
1/50
Total
0/1
 
Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). (If this is not possible, enter DNE in any single blank.)
0−110
5105
1−130
011−1
Incorrect: Your answer is incorrect.

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10. 2/4 points  |  Previous Answers PooleLinAlg4 3.5.035.EP. My Notes
Question Part
Points
Submissions Used
1 2 3 4
0/1 0/1 1/1 1/1
2/50 2/50 1/50 1/50
Total
2/4
 
Consider the following matrix.
A = 
30−3
222
Find bases for row(A) and col(A).
row(A)

Incorrect: Your answer is incorrect.
column(A)

Incorrect: Your answer is incorrect.
Give the rank and the nullity of the matrix.
rank(A) = Correct: Your answer is correct. nullity(A) = Correct: Your answer is correct.
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