** 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.

- Section 1.6, Problems 1, 7, 9, 10 (unless you did them last week.) Also Section 1.6, Problems 29, 33 and 35 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, $\mathcal{F}(F,F)=\mathrm{Pol}_{q-1}(F)$. (Here $\mathrm{Pol}_n(F)$ is not to be confused with $P_n(F)$ from the textbook.)

- The midterm test takes place on Tuesday, October 17, 11:10-1:00 in the Examination Centre.
- Please bring your student ID !!
- It's `no tools allowed' -- especially no electronic gadgets.
- It's a `crowdmark' exams, which means that we'll scan your solutions,
and grade and return online. It is hence ok (perhaps even recommendable) to use a pencil, but if so, please use a
**dark**pencil since otherwise it won't come through on the scan. - Material covered: Anything covered in this course until then. In terms of the textbook chapters, this is Appendices A-D and Chapters 1.1-1.6. We didn't manage to cover Lagrange interpolation before the midterm, so it won't be on teh midterm.
- See Blackboard, under Course Materials, for last year's midterm. (With solutions.)
- There will be additional TA office hours before the test; for information see the recent Blackboard announcement.