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PreTeXt Sample Book: Abstract Algebra (SAMPLE ONLY)

Thomas W. Judson, Isaac Newton (Editor)

Appendix B Hints and Answers to Selected Odd Exercises

View Source for solutions
<solutions divisional="hint answer solution" admit="odd">
  <title>Hints and Answers to Selected Odd Exercises</title>
</solutions>

II Algebra
3 Runestone Testing
3.8 True/False Exercises

3.8.1. True/False.

Hint.
View Source for hint
<hint>
  <p>
    <m>P_n</m>, the vector space of polynomials with degree at most <m>n</m>,
    has dimension <m>n+1</m> by <xref ref="theorem-exponent-laws" />. [Cross-reference is just a demo,
    content is not relevant.] What happens if we relax the defintion and remove the parameter <m>n</m>?
  </p>
</hint>
\(P_n\text{,}\) the vector space of polynomials with degree at most \(n\text{,}\) has dimension \(n+1\) by Theorem 1.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter \(n\text{?}\)

3.9 Multiple Choice Exercises

3.9.1. Multiple-Choice, Not Randomized, One Answer.

Hint 1.
View Source for hint
<hint>
  <p>
    What did you see last time you went driving?
  </p>
</hint>
What did you see last time you went driving?
Hint 2.
View Source for hint
<hint>
  <p>
    Maybe go out for a drive?
  </p>
</hint>
Maybe go out for a drive?

3.9.3. Multiple-Choice, Randomized, One Answer.

Hint 1.
View Source for hint
<hint>
  <p>
    What did you see last time you went driving?
  </p>
</hint>
What did you see last time you went driving?
Hint 2.
View Source for hint
<hint>
  <p>
    Maybe go out for a drive?
  </p>
</hint>
Maybe go out for a drive?

3.9.5. Mathematical Multiple-Choice, Not Randomized, Multiple Answers.

Hint.
View Source for hint
<hint>
  <p>
    You can take a derivative on any one of the choices to see if it is correct or not,
    rather than using techniques of integration to find
    <em>a single</em> correct answer.
  </p>
</hint>
You can take a derivative on any one of the choices to see if it is correct or not, rather than using techniques of integration to find a single correct answer.

3.10 Parsons Exercises

3.10.1. Parsons Problem, Mathematical Proof.

Hint.
View Source for hint
<hint>
  <p>
    Dorothy will not be much help with this proof.
  </p>
</hint>
Dorothy will not be much help with this proof.

3.12 Matching Exercises

3.12.3. Matching Problem, Linear Algebra.

Hint.
View Source for hint
<hint>
  <p>
    For openers, a basis for a subspace must be a
    <em>subset</em> of the subspace.
  </p>
</hint>
For openers, a basis for a subspace must be a subset of the subspace.

3.13 Clickable Area Exercises

3.13.3. Clickable Areas, Text in a Table.

Hint.
View Source for hint
<hint>
  <p>
    Python boolean variables begin with capital latters.
  </p>
</hint>
Python boolean variables begin with capital latters.

3.18 Fill-In Exercises

3.18.11. Fill-In, Dynamic Math with Formulas as Answers.

3.19 Hodgepodge

3.19.1. With Tasks in an Exercises Division.

3.19.1.a True/False.
Hint.
View Source for hint
<hint>
  <p>
    <m>P_n</m>, the vector space of polynomials with degree at most <m>n</m>,
    has dimension <m>n+1</m> by <xref ref="theorem-exponent-laws" />. [Cross-reference is just a demo,
    content is not relevant.] What happens if we relax the defintion and remove the parameter <m>n</m>?
  </p>
</hint>
\(P_n\text{,}\) the vector space of polynomials with degree at most \(n\text{,}\) has dimension \(n+1\) by Theorem 1.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter \(n\text{?}\)

3.20 Exercises that are Timed

Timed Exercises

3.20.1. True/False.
Hint.
View Source for hint
<hint>
  <p>
    <m>P_n</m>, the vector space of polynomials with degree at most <m>n</m>,
    has dimension <m>n+1</m> by <xref ref="theorem-exponent-laws" />. [Cross-reference is just a demo,
    content is not relevant.] What happens if we relax the defintion and remove the parameter <m>n</m>?
  </p>
</hint>
\(P_n\text{,}\) the vector space of polynomials with degree at most \(n\text{,}\) has dimension \(n+1\) by Theorem 1.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter \(n\text{?}\)

3.27 Group Exercises

3.27.1. Multiple-Choice, Not Randomized, One Answer.

Hint 1.
View Source for hint
<hint>
  <p>
    What did you see last time you went driving?
  </p>
</hint>
What did you see last time you went driving?
Hint 2.
View Source for hint
<hint>
  <p>
    Maybe go out for a drive?
  </p>
</hint>
Maybe go out for a drive?
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