Math 361, Spring 2020, Assignment 13

Read:

1. Section 30.
2. Section 31.

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

1. Vector space (over a field $F$).
2. Linear combination (from a subset $S$ of a vector space $V$; also known as an $F$-linear combination when the field of scalars might be in doubt).
3. Span (of a subset $S$ of a vector space $V$; a.k.a. $F$-span).
4. Linear relation (among elements of a subset $S$ of a vector space $V$; a.k.a. $F$-linear relation).
5. Trivial linear relation.
6. Linearly independent (set $S$ in a vector space $V$; a.k.a. $F$-linearly independent).
7. Basis (for a vector space $V$; a.k.a. $F$-basis).
8. $\mathrm{dim}_FV$ (the dimension of $V$ over $F$, a.k.a. the $F$-dimension of $V$).
9. $[E:F]$ (the degree of the field extension $F\rightarrow E$).
10. Homomorphism (of vector spaces over $F$; a.k.a. $F$-linear transformation).
11. Isomorphism (of vector spaces over $F$).

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

1. Theorem regarding the existence of bases.
2. Theorem relating the cardinality of any two bases for the same space $V$ over the same field $F$.
3. Theorem concerning the extension of linearly independent sets to bases.
4. Theorem concerning the refinement of spanning sets to bases.
5. The Dimension Formula.
6. Procedure to find an $F$-basis for $E$, given a nested extension $F\rightarrow K\rightarrow E$, an $F$-basis for $K$, and a $K$-basis for $E$.
7. Lagrange-like corollary of the Dimension Formula, relating the degree of a subextension to the degree of the parent extension.
8. Formula for $\left[F[x]/\left\langle m\right\rangle:F\right]$.
9. Procedure to find an $F$-basis for $F[x]/\left\langle m\right\rangle$.

Solve the following problems:

1. Section 30, problems 1, 2, 5, 6, and 9.
2. Section 31, problems 1, 3, 5, 7, 22, 23, and 24.
3. (Math 260 refresher) Consider the polynomials $f_1,f_2,f_3\in\mathbb{Z}_5[x]$ defined by $f_1=x^2+2x+1, f_2=3x+1, f_3=x^2+3x+3$. Determine whether the set $\{f_1,f_2,f_3\}$ is linearly independent over $\mathbb{Z}_5$. (Hint: all of these polynomials lie in the three-dimensional subspace $V$ of $\mathbb{Z}_5[x]$ with basis $\{1,x,x^2\}$. Extraction of coordinates with respect to this basis sets up an isomorphism $V\simeq\mathbb{Z}_5^3$. Push the problem through this isomorphism, then use a standard Math 260 algorithm to determine whether the resulting triple of column vectors is linearly independent. In the end you will answer the question by executing the Gauss-Jordan algorithm on a certain $3\times3$ matrix with entries in $\mathbb{Z}_5$.)

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