Hello everyone

I'm just putting together a first pass at the wording for P1385, A proposal
to add linear algebra support to the C++ standard library
<https://www.open-std.org/jtc1/sc22/wg21/docs/papers/2022/p1385r7.pdf>. If
you look at the latest revision you will infer that at Kona in November and
at Issaquah in February I addressed SG6 and LEWG about withdrawing the
vector class entirely and simply offering a matrix class, where a vector is
a special case of a matrix, with a single row or column. There were no
objections to this approach.

While there were no objections raised in the meeting, others have come in,
and I want to use the reflectors to gather opinion about the matter. The
heart of the problem is: what does the vector product signify? Is it an
inner or outer product? Is vector orientation significant?

With my mathematician's hat on, multiplying a row vector by a column vector
is an inner product, yielding a scalar value if both vectors have the same
number of elements. Appearing much more rarely, multiplying a column vector
by a row vector is an outer product yielding a square matrix.

However, in the domains where I most make use of linear algebra, what is
more significant is that the vector-matrix product treats the vector as a
row vector, the matrix-vector product treats the vector as a column vector,
and the vector-vector product treats the operation as an inner product.
Orientation is irrelevant, and transposing vectors is a tedious waste of
time.

The proposal is, under the hood, a way of bringing together element types,
element conversions, extents and kernels into a single type, borrowing
parts from mdspan. All of these items are template parameters. My current
intent is to restore row and column vectors to the proposal, with
additional motivation, and supply Orientation-Significant Kernels. I intend
to offer Orientation-Agnostic Kernels in a subsequent paper (unless someone
beats me to it, which you are all really, REALLY welcome to do) to enable
this use-case as standard, allowing users to decide whether orientation is
significant. Example code to follow in the fullness of time.

Speaking of which, time is tight: I have a fortnight until the Varna paper
deadline, so I would appreciate any commentary sooner rather than later.

Cheers,
G