Week 4: October 3-7, 2016
Course Change Date: Monday, October 3, is the last day to change from MAT240 to MAT223 without `academic penalty'. Click
this link for
information on how to change.
Required reading: Sections 1.5, 1.6.
By the way, the `required readings'
really are `required'. In particular, some assignment problems may rely on material from those sections, even if it wasn't covered in the lecture.
Assignment #3 has been posted, and is due on Friday, October 7. Please send me an email, in case you did not receive the Crowdmark email. Solutions to #2 have been posted on Blackboard.
Some recent FAQ's regarding the Crowdmark assignments:
- Q: Should I write my name and student number on the submitted pages? A: It is not necessary; Crowdmark links the submisson to your email address. In fact, we prefer if not, because the grader won't know who you are.
- Q: I didn't receive my Crowdmark email, or lost the link. What to do? A: Send me an email; I can re-send your personal link.
-
Q: I have trouble uploading, what to do? A: If absolutely necessary, send
me your work as an email attachment.
Additional homework (do not hand in).
- Section 1.5, Problems 1, 2, 3, 9, 10, 15, 18. Section 1.6, Problems 1, 7, 9, 10
- 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?
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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 $A$. For instance, some subsets of $\mathbb{R}$:
$$\# \emptyset =0,\ \ \#\{0\}=1,\ \ \#\mathbb{Z}=\infty,\ \ \
\#\{-1,1\}=2.$$
(Note that in $\mathbb{Z}_2$, we would have $\#\{-1,1\}=1$, because $1=-1$ in the field $\mathbb{Z}_2$ -- and repetitions don't matter if one enumerates elements of a set.)
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.