SB Hodgepodge

Exercises 5.19 Hodgepodge

1. With Tasks in an Exercises Division.

Structured with task, recycled earlier from earlier, to make sure that the tasks do not get counted as Runestone reading activities (since they are inside an <exercise> inside of an <exercises> division.

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(a) True/False.

Every vector space has finite dimension.

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

Hint.

P_{n},

the vector space of polynomials with degree at most

n,

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 ?

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(b)

Explain your reasoning in the previous question.

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