AppendixCHints and Answers to Selected Even Exercises
1Preliminaries 1.4Exercises
Warm-up
1.4.2.
Hint.
(a) ; (d) .
1.4.6.
Hint.
If , then either or . Thus, and . Hence, . Therefore, . Conversely, if , then and . Thus, or is in both and . So and therefore . Hence, .
1.4.10.
Hint.
.
1.4.14.
Hint.
.
More Exercises
1.4.18.
Hint.
(a) is one-to-one but not onto. . (c) is neither one-to-one nor onto. .
1.4.20.
Hint.
(a) .
1.4.22.
Hint.
(a) Let . Then . Thus, and , so is one-to-one. (b) Let , then for some . Since , is onto.
1.4.24.
Hint.
(a) Let . Then there exists an such that . Hence, or . Therefore, . Consequently, . Conversely, if , then or . Hence, there exists an or there exists an such that . Thus, there exists an such that . Therefore, , and .
1.4.28.
Hint.
Let and define if .
3Groups 3.5Exercises
3.5.2.
Hint.
(a) Not a group; (c) a group.
3.5.6.
Hint.
3.5.8.
Hint.
Pick two matrices. Almost any pair will work.
3.5.16.
Hint.
Look at the symmetry group of an equilateral triangle or a square.
3.5.18.
Hint.
Let
be in . All of the s must be distinct. There are ways to choose , ways to choose ,, 2 ways to choose , and only one way to choose . Therefore, we can form in ways.
3.5.46.
Hint.
Look at .
3.5.56.
Answer.
3.5.58.
Answer.
3.5.60.
3.5.60.a
Answer.
3.5.60.b
3.5.60.b.i
Answer.
3.5.60.b.ii
Answer.
5Runestone Testing 5.9Multiple Choice Exercises
5.9.4.Multiple-Choice, Randomized, One Answer.
Hint1.
What did you see last time you went driving?
Hint2.
Maybe go out for a drive?
5.9.6.Mathematical Multiple-Choice, Not Randomized, Multiple Answers.
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.
Do you really need a hint? Carefully reread the question.
5.18.10.Fill-In, Dynamic Math with Simple Numerical Answer.
5.18.12.Fill-In, Dynamic Math with Interdependent Formula Checking.
5.19Hodgepodge
5.19.2.With Tasks in an Exercises Division.
5.19.2.aTrue/False.
Hint.
, the vector space of polynomials with degree at most , has dimension by Theorem 3.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter ?
5.20Exercises that are Timed
Timed Exercises
5.20.2.Multiple-Choice, Not Randomized, One Answer.