Math 361, Spring 2017, Assignment 9

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

1. Vector space (over a field $F$).
2. Subspace (of a vector space).
3. Subspace generated (a.k.a. spanned) by a set $S$.
4. Linear relation (in a set $S$).
5. Linearly independent.
6. Basis.
7. $\mathrm{dim}_F V$ (a.k.a. the dimension of $V$ over $F$).
8. $[E:F]$ (a.k.a. the degree of $E$, regarded as an extension of $F$).

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

1. Theorem concerning the existence of bases.
2. Theorem relating the cardinalities of two bases for the same vector space.
3. Theorem relating bases to unique expansion of vectors as linear combinations.
4. Dimension Formula.
5. Corollary of the Dimension Formula analogous to Lagrange's Theorem in group theory.
6. Corollary concerning subextensions of field extensions of prime degree.
7. Theorem characterizing algebraic elements in terms of finite-dimensional subextensions.
8. Theorem concerning sums, products, and inverses of algebraic elements.

Solve the following problems:

1. Section 30, problems 1, 4, 5, 6, 7, and 8.
2. Section 31, problems 1, 3, 5, 7, 22, 23, 24, 25, and 29.

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Questions

The first homework question talks about bases for $\mathbb{R}^2$ no two of which have a vector in common. I am not sure where to start to find the three bases. Is the next several question ask a given basis for the indicated vector space over the field. Not sure where to start, with those questions. Where do I start to solve the problems?

Any two vectors in $\mathbb{R}^2$, neither of which is a scalar multiple of the other, will form a basis for $\mathbb{R}^2$. So, for example, $\begin{bmatrix}1\\1\end{bmatrix}$ and $\begin{bmatrix}1\\2\end{bmatrix}$ will form a basis.

Any simple algebraic extension has a "standard basis" $\{1,\alpha,\dots,\alpha^{d-1}\}$ where $\alpha$ is the generator and $d$ is the degree of $\alpha$ over the base field. So, for example, $\mathbb{Q}(\sqrt[5]{2})$ has $\mathbb{Q}$-basis $\{1, \sqrt[5]{2}, (\sqrt[5]{2})^2, (\sqrt[5]{2})^3, (\sqrt[5]{2})^4\}$. -Steven.Jackson (talk) 10:28, 4 April 2017 (EDT)

I have a problem with question 22 section 31. The question is: Let $(a+bi) \in \mathbb{C}$ where $a, b \in \mathbb{R}$ and $b \neq 0.$ Show that $\mathbb{C} =\mathbb{R}(a+bi)$.(22 is not in the back of the book as an answer) Thanks again for the help.

The simplest way to do this is to compute the dimension of $\mathbb{R}(a+bi)$, regarded as a vector space over $\mathbb{R}$. It is a subextension of $\mathbb{C}$, which has dimension two, so its dimension is either $1$ or $2$. In the first case, the subextension is trivial, contradicting the fact that $b\neq0$ (hence $a+bi\not\in\mathbb{R}$). So the dimension is two, and $\mathbb{R}(a+bi)$ is all of $\mathbb{C}$. -Steven.Jackson (talk) 21:20, 9 April 2017 (EDT)