this is the limit

November 20, 2012 in Notations, Rambles

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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