Math 361, Spring 2015, Assignment 7

Carefully define the following terms, then give one example and one non-example of each:

1. Finitely generated extension.
2. Vector space (over a field $F$).
3. $F$-linear transformation (between two vector spaces).
4. $F$-linear combination (of elements of a set $S$).
5. $F$-span (of a set $S$).
6. $F$-spanning set.
7. $F$-linearly independent set.
8. $F$-basis.
9. $\mathrm{dim}_FV$.
10. Coordinate vector (of $\vec{v}\in V$ with respect to an ordered basis $\mathcal{B}\subseteq V$).
11. Matrix (of a linear transformation $T:V\rightarrow W$ with respect to a pair of ordered bases $\mathcal{B}\subseteq V$ and $\mathcal{C}\subseteq W$).

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning extension of linearly independent sets to bases.
2. Theorem concerning refinement of spanning sets to bases.
3. Theorem relating the cardinalities of various bases for the same vector space.

Solve the following problems:

1. Section 30, problems 1, 3, 4, and 5.
2. Suppose that $F\stackrel{\iota}{\rightarrow} E$ is a field extension. Show that the scalar multiplication $a\cdot e = \iota(a)e$ turns $E$ into a vector space over $F$. (Here the multiplication on the right hand side is carried out in the field $E$.)
3. Regard $\mathbb{C}$ as an extension of $\mathbb{R}$ in the usual way. Calculate $\dim_{\mathbb{R}}\mathbb{C}$.
4. Regard $\mathbb{R}$ as an extension of $\mathbb{Q}$ in the usual way. Calculate $\dim_{\mathbb{Q}}\mathbb{R}$. (Warning: this is actually a fairly hard exercise in set theory. But at least see if you can guess the answer.)

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