## Archive for November, 2012

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!

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* ()

with its mindbendingly-wrong “plain english”

equivalent(s). the perfectly-correct

(and altogether-necessary) symbolism

(“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,

.

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 ;

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 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 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:

and , 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.