##### Week 5: October 9-13, 2017.
• Homework #4 is due 11pm on Friday, October 13
• Required reading: Chapters 1.5, 1.6
• Bases and dimension. Replacement lemma. Examples and applications: Lagrange polynomials, the dimension of a vector space over a finite field. See also here. The formula for the dimension of a sum of subspaces, $$\dim(W_1+W_2)+\dim(W_1\cap W_2)=\dim(W_1)+\dim(W_2).$$
• From now on, we'll make use of the following notations: If $A,B$ are sets, we denote $A\backslash B = \{x\in A|\ \ x\not\in B\}.$ The symbol $\# A$ will denote the cardinality of $A$, that is, the number of elements in $A$.
• 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.

• Additional homework (do not hand in).
• 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.)
• Just for fun: A famous linear algebra puzzle: In a town with $n$ inhabitants, there are $N$ clubs. Each club has an odd number of members, and for any two distinct clubs there is an even number of common members. Prove that $n\ge N$. Hint: Work with the field $\mathbb{Z}_2$, and use facts about bases and dimension. Solution.
• It's no tools allowed' -- especially no electronic gadgets.