Am Di., 13. Okt. 2020 um 21:45 Uhr schrieb Arthur A. Gleckler
<xxxxxx@speechcode.com>:
>
> On Tue, Oct 13, 2020 at 10:44 AM Marc Nieper-Wißkirchen <xxxxxx@nieper-wisskirchen.de> wrote:
>
>>
>> Speaking of linguistics, the word vector comes from Latin, meaning
>> carrier in the sense what it needs to have something carried from
>> point A to B. This is still very much the meaning in the natural
>> sciences. An ordinary Scheme vector fits this picture very well (as it
>> can be considered as a coordinate vector in some generalized
>> coordinate system).
>
>
> I follow your explanation of etymology until you mention coordinate systems.  There's no need to talk about coordinate systems to fit the picture of a carrier.  A vector in the sense of a carrier is just a thing that carries other things.

Indeed, for the abstract notion of a vector coordinate systems aren't
necessary and the abstract algebra of vectors works without. In
astronomy, where this use of the Latin word seems to originate, the
difference between two points is just one arrow in space but a priori,
not a thing made of numbers.

Nevertheless, when we impose a coordinate system on our space (which
we usually do to do actual calculations) we can represent a vector by
its coordinate vector because a coordinate system identifies the
abstract space with, in the case of classical astronomy, R^3. By
allowing arbitrary dimensions, a coordinate vector, therefore, matches
a Scheme vector with real numbers as entries quite well.

The next step is Lagrangian dynamics where we replace the usual space
coordinates (and their velocities) by generalized coordinates. These
generalized coordinates do not have to have a unit of length anymore
so when I represent a vector in such a generalized coordinate system,
its entries do not have to be of a homogeneous type anymore. So, in
some sense, we have arrived at a general Scheme vector: a fixed-length
tuple of values.