Week 4: October 26, 2017
The front row seat, right next to the center aisle, will from now on be reserved for one of your classmates who needs this arrangement.
Required reading: Chapters 1.3, 1.4, 1.5.
Vector spaces: Basic properties, subspaces, intersections and sums of
subspaces, linear combinations, linear dependence and independence.
Homework #3 is due 11pm on Friday, October 6
Additional homework (not to be handed in):
 Prove the `basic properties' of vector spaces mentioned in class:
 $x+y=x'+y\ \ \Rightarrow\ \ x=x'$,
 $a x=a x',\ a\neq 0\ \ \Rightarrow\ \
x=x'$
 $a x= a' x,\ x\neq 0 \ \Rightarrow\ \
a=a'$
 $0\cdot x=0$ for all $x\in V$,
 $ax=0\ \ \Rightarrow\ \ a=0 \mbox{ or } x=0$
 The empty set is not a vector space. Why?

Section 1.4, Problems 2,3,12,13,14,15. (At this point
we won't cover much along the lines of problem 2,3, but we'll do it soon, more systematically. It's good to get some practice with this now .)
 Section 1.5, Problems 1, 2, 3, 9, 10, 15, 18.
 Let $V$ be the vector space over $F$, consisting of all infinite sequences
$(a_1,a_2,\ldots)$ with $a_i\in F$. Let
\[ S=\{(1,0,0,0,\ldots),\ (0,1,0,0,\ldots),\ (0,0,1,0,\ldots),\ \ldots\}.\]
Show that $S$ is linearly independent.
What is $\operatorname{span}(S)$? Can you find another linearly independent subset
$T\subset V$, also with infinitely many elements, and such that $\operatorname{span}(S)\cap \operatorname{span}(T)=\{0\}$ In fact, such that $S\cup T$ is still linearly independent?
Just for fun: A famous linear algebra puzzle (clever but not easy): 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.