Archive for November, 2012

this is the limit

so i’m grading linear algebra like i usually do.

where we need to speak… or anyway, to read
and write… frequently about *column vectors*.

much the usual thing (for example) in defining
a linear transformation (called F, say)
on “real three-space” (so F: R^3 —> R^3)
is to *consider R^3 as the space of real-valued
column 3-vectors* and then to supply a matrix
(called [F], say; [F] will be a 3-by-3
matrix in this context) such that
F([a, b, c]^T) = [F][a, b, c]^T.

the “caret-T” here denotes “transpose”…
the idea is that one has something like
[a, b, c]^T=
[
a
b
c
]
;
in laypersons’ language, the transpose
symbol ^T tells us to turn our old rows
into the new columns (which simultaneously
turns our old columns into the new rows).

in the language of the widely-used
TI-* calculator line, one has
[a, b, c]^T = [[a][b][c]]…
and this is starting to look
better and better to me right
in here.

but what one really seems to *want*
here is a quick-and-dirty notation
for expressing (what we will still
continue to *speak* of as)
column-vectors, as rows.

and i’ve noticed student papers using
< x, y, z > = [x, y, z]^T.
this looks like a real useful convention
to me and i’ve adopted it for my own use
until further notice.

angle-bracketed vectors have been useful
to me before. mostly, i think, in the context
of “sequences & series” typically dealt with
in about calc 2 or 3.

LANGLE x_0, x_1, x_2, … RANGLE
(i.e., < x_0, x_1, x_2, … >
… “angle brackets” are special characters
in HTML and so i prefer to avoid ’em)…
in either notation…
represents sequence of objects
LANGLE x_n RANGLE
(which is of course *not* the same
as the *set* R={ x_n } = {x_0, x_1, x_2, …}
[the set of values taken by the function
f(n) = x_n
on the natural numbers NATS:={0, 1, 2, …}]).

this usage actually *extends* a usage,
ideally introduced and maintained earlier
in a given presentation (class or text or
who knows maybe someday even both at once)
using angle-brackets for (finite-dimensional)
*vectors*.

LANGLE 3, 4 RANGLE
now represents the vector that, ideally,
we would represent in some other part of
our presentation as [3,4]^T…
a *column vector*.

i remark here that meanwhile
(3,4)
represents a so-called “point”
in “the x-y plane”… an entirely different
(though closely related) object.

we pause here and take a deep breath.

i’ve pointed at two distinctions:
sequences-as-opposed-to-sets and
vectors-as-opposed-to-points.

many textbooks… and many instructors…
are *very sloppy* about keeping these
(and many other suchlike) distinctions clear.

the *notations* used in *distinguishing*
the distinct situations in each case were
“delimiters”: pairs of opening-and-closing
symbols used to mark off pieces of code
meant to be handled as single objects.

delimiting conventions are vital even in
ordinary literacy (“you see? he” sa)i(d.
and i claim they’re all the more so in maths
(since we get fewer and weaker “context clues”
when the code gets munged [as in the example]).

and students’ll just leave ’em out altogether
if they think… or god help me, know… they
can get away with it. failing that, choose
randomly (itt…oghm,k…).

failing that, “well, you *know*
what i *meant* was”…

too late, too late. here endeth the sermon.

in our next episode of “who stole my infrastructure?”:
the dot product. when they came for the opening-apostrophe,
i pleaded and begged. when they came for the
sign-of-intersection i raved incoherently.
never had a chance, no hope, no hope. doom doom doom.
can somebody pick up the torch, here?
i don’t think i can go on much longer.

vlorbik on punctuation for the twenty-twelve.
more clarity!

midterm report

at the foundation of (an earlier version of)
this blog i ranted and rambled about
a tendency on the part of (lower-division
college math) textbooks to hamper the work
of the teacher by (deliberately!) suppressing
correct technical language.

of course things have continued to deteriorate.
but, by some miracle, i’m still earning
the random crust of bread by helping students
learn to *read* these ever-more-horribly flawed
documents. so far so good, then, i suppose.

anyhow, i’m grading linear algebra again
right in here (nothing *but* linear algebra
for something like a *year* now)…
and i’ve only recently become vividly aware
that this “tendency” has penetrated deeply into
the textbooks at this “higher level of the game”.

specifically, i hereby announce that some
satanic force has somehow (even here) replaced
*the sign of set-membership* (\in)
with its mindbendingly-wrong “plain english”
equivalent(s). the perfectly-correct
(and altogether-necessary) symbolism
x \in S (“x is an element of S”)
is now to be replaced, by the edict of
invisible (and mostly unimaginable)
entities, with “x is in S”.

[
this is a good place to skip ahead.

i’m going to geek out slightly here.
you *don’t* have to be an adept to follow.
i’m hoping to make a point that can be
at least *partly* understood by math laity.

is the set of natural numbers (more here
much my most popular post here and probably
my best-read production of all time) “in” the set
of real numbers? loosely, yes. more precisely,
{\Bbb N} \subset {\Bbb R}.

i can easily imagine myself talking to, say, another
teacher about, say, some “property” (like commutativity-
-of-addition; x+y=y+x [for all x & y]) that applies
in the natural numbers. “how do we *know* it applies?”,
i might say. and the answer might come: “because
the naturals are in the reals, and the *reals* enjoy
the property of commutativity”. “good answer!” i would
then reply, and move on to whatever i *really* wanted
to talk about.

again. are all possible probabilities in the reals?
well, yeah, (duh)! in “code”, one has [0, 1] \subset {\Bbb R};
rephrasing, “all the numbers from zero to one (inclusive)
are in the reals” (but also, more precisely,
“the [closed] unit interval is a subset of the set of real numbers”)…
so. now i’m talking to some grad-school dropout (say): “is
the-interval-from-zero-to-one in the reals?” she asks;
“heck, yes” say i, and we get on with whatever we’re really doing.

is “pi” in the real numbers? sure! \pi \in {\Bbb R}.
“pi is an element of the reals”.

but wait! being-a-subset is an *entirely different* relation
from being-an-element! pi is simply *not* “in” the reals
in the same way that \Bbb N is!

who cares? well, me and a few hundred thousand others or so.
if *you* don’t care? well, that’s why i invited you
to skip this part! read on!
]

the biggest problem from a practical standpoint
(if “how can we make this material better understood”
is a practical question) is simply that students
*hate writing* and at *every opportunity* will
replace “plain english” with (typically very
ill-understood) bits-and-pieces.

*nobody*… no student, no lecturer, no pro
mathematician… is going to write out the phrase
“is a real number” a whole lot more than
twenty or thirty times (in a given sitting-down)
without wanting *some* abbreviation for that
phrase.

and likewise for “is a subset of”… indeed,
*any* sufficiently common phrase *begs for
abbreviation* even in “plain english”

so there it is. mathophiles also… in some sense…
“hate writing”. anyhow, we *love abbreviating*.

“algebra is the science of equations”, i once
heard someone say (explicitly repeating something
he had learned “by rote” from a public-school
teacher during his own schooling… it went on
for another few lines but i didn’t learn that
part from listening to this guy say it three
or four times that one night). and i consider
this to’ve been very well said.

so i’ll hope to return to it.

but first. this history of elementary algebra
at w’edia summarizes the standard dogma of its subject (as i
understand it) well. the evolution from “rhetorical” algebra
(describe *everything* in plain-language words) through “syncopated”
algebra (where “shorthand” symbols [many still common today]
began to replace the most common techical terms… but the
actual *reasoning* was still natural-language based [and so,
by contemporary standards, “informal”]) into
“symbolic algebra” (the “science of equations” as we know it
today: a study of “formal” properties of [carefully-defined!]
symbolic “objects” [“variables” and “equations”, for example]).

what w’edia *doesn’t* make much of… but what matters to me
a great deal… is that the emergence of algebra pretty
closely *coincides* in (so-called) *western* history with
the (so-called) renaissance and the (no sneerword necessary)
scientific revolution.

“modern times”, then, *began* when certain humans (*finally*!)
figured out how collections of symbols-on-paper (representing
certain abstractly-defined-objects), produced according to
various “rules”, could be interpreted to reveal previously-obscure
*laws of nature* (so-called). this *was* the “scientific revolution”.

and how does it work? *equations* is how.
so one of the *first* things in understanding
what’s going on the contemporary philosophical
environment is to find out *what an equation is*
(for all literate people): “the equality meaning
of the equals sign”
.

when we’re being sloppy, we can confuse “=” with “is”…
but when we actually get to work *using* equations,
we have to *much more precise* to get any value from
the procedure at all. plain-english “is” is *always*
in some sense metaphorical (except in empty utterances
like “it is what it is”)… whereas the equal-sign
rightly-used is as far away from metaphor as we know
how to get.

how does algebra work? (equations and *what* else?)
by *the method of substitution* is how.
“in a context such that A=B is taken as ‘true’,
a properly-written piece of code
including (the symbol) B
*does not change its truth-value*
when (the symbol) A is *substituted* for B.”

this “method” *characterises* algebra.
i first became aware of some its awesome power
in about seventh grade.

and *what* else? “doing the same thing to
both sides of an equation”. and what else?
that’s about it. that’s algebra. the rest
is commentary.

now, for *set theory*, two of the main ideas
are caught up in set-inclusion and set-containment:
\pi \in \Bbb R and {\Bbb N} \subset \Bbb R, for example.

and one must be every bit as careful in the use
of these symbols when studying sets as one must be
in the use of the sign-of-equality in studying,
say, polynomial equations (i.e., pretty much,
in algebra).

but about forty percent of the class already don’t
take *equations* seriously. and they’re morally
certain that “sets” are meaningless traps designed
to distract them from “how do i get the *answer*?”.

and the textbook does a great deal to encourage them
to *maintain* this view.

and i don’t like it. please stop.