PreTeXt Sample Book: Abstract Algebra (SAMPLE ONLY)

Section5.14Exercises that are Timed

This is a section that merely explains and holds an <exercises> division, which will be at the level of a <subsection>. There is a @time-limit attribute on <exercises>, set to the value 10, which implies (a) the collection of (two) exercises is a “timed exam” when hosted on Runestone, and (b) a student will have 10 minutes to complete the collection.
Showing results, showing feedback, displaying a timer, and allowing pausing are all enabled by default. To disable any of these features, set the corresponding attributes on the <exercises> division, @results, @feedback, @timer, @pause, to the value no. As a test, we have turned off pausing. Don't panic!
Of course, if you are not viewing this while online and hosted on a Runestone server, then these exercises will not look any different than in other places.
(Since this is an unstructured division, the number of the <exercises> is not displayed when born. It does have a number, which is the same as the enclosing <section>. To wit: Section 5.14 versus Exercises 5.14.)

ExercisesTimed Exercises

You have 10 minutes to do these exercises when hosted online on a Runestone server.

1.True/False.

Every vector space has finite dimension.
• True.

• The vector space of all polynomials with finite degree has a basis, $$B = \{1,x,x^2,x^3,\dots\}\text{,}$$ which is infinte.
• False.

• The vector space of all polynomials with finite degree has a basis, $$B = \{1,x,x^2,x^3,\dots\}\text{,}$$ which is infinte.
Hint.
$$P_n\text{,}$$ the vector space of polynomials with degree at most $$n\text{,}$$ has dimension $$n+1$$ 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 $$n\text{?}$$