### 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!

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