Week 5: October 10-14, 2016.

About the upcoming midterm exam: click here.

Required reading: Section 1.6.; also take a first look at 2.1

Assignment #4 is due on Friday, October 14, 11 pm. Please send me an email, in case you did not receive the Crowdmark email.

Please be sure to `validate' your submissions, to make sure they really went through. The images shown in the validation view are exactly the same as those shown to the grading team.

Additional homework (do not hand in).

Section 1.6, Problems 1, 7, 9, 10 (unless you did them last week.) Section 2.1, Problems 1, 2, 3, 21
Let $F$ be a finite field with $q$ elements. Show that every function $f\in\mathcal{F}(F,F)$ is uniquely a polynomial of degree $\le q-1$ with coefficients in $F$. That is, $\mathcal{F}(F,F)=\mathcal{P}_{q-1}(F)$.

In class, we used a different (but equivalent) formulation of Theorem 1.10 of the textbook:

Replacement Lemma : Let $V$ be a vector space, and $S$ a subset generating $V$, i.e. $\operatorname{span}(S)=V$. Suppose $v_1,\ldots,v_m\in V$ are linearly independent vectors. Then there exist distinct vectors $u_1,\ldots,u_m\in S$ such that the subset $$ (S\backslash \{u_1,\ldots,u_m\})\cup \{v_1,\ldots,v_m\}$$ spans V.