Axler Chapter 1 Exercises — Studies in Algebra and Group Theory

N16 Axler Chapter 1 Exercises

In progress: these notes stop partway.

Subspaces (Axler 1C, Exercise 1)

Exercise N16.1. For each of the following subsets of , determine whether it is a subspace of :

(a)

Exercise N16.2 (Problem 1). Find with , i.e.

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Comparing real and imaginary parts: From the first equation , so (for ) . Substituting into the second, (If then , and , which is the same formula.)

Exercise N16.3 (Problem 2). Show that .

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Exercise N16.4 (Problem 3). Find two distinct square roots of .

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Exercise N16.5 (Problem 4). Show that for all .

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Given that this is a field this is not needed, but okay. Let be and be . So then, via simple algebra, holds for both if it holds for .

Exercise N16.6 (Problems 5 and 6). 5: for all . 6: .

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SAME! Fuck it, I’ll do it anyways.

#5. We use:

Theorem N16.7. for all .

Let , , , for , so . Then both sides sum to

#6. : the same follows from .

Exercise N16.8 (Problem 7). Show that for every there exists a unique with .

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Assume that there exists such that