Linearity: the extended consequences of addition

When quantities are amenable to addition, one can use repeated addition to build up a notion of scaling by the positive naturals. In so far as such scalings are invertible, one may construe the inverses (division by positive naturals) as scaling by reciprocals of naturals; and composites of natural scalings and reciprocal scalings as scalings by rationals (general fractions). [Where the quantities involved form a continuum, we can look to extend scalings to a topologically complete continuum, having the form of orthodoxy's real number continuum. I would like, however, to avoid building the reals per se as far as I can, except in so far as they arise naturally as scalings.] Orthodoxy describes such structures in terms of vector spaces.

Orthodoxy's vector spaces are, by definition, additive groups: this implies (at least) that they have to have negatives; yet some of the quantities with which physics wishes to deal (e.g. temperatures and masses) are intrinsically positive, with zero only as a notional or theoretical limit, yet support meaningful addition – and scaling by positive values. Orthodoxy also (mirroring its general progression from sets to functions) begins by defining a vector space, then introduces linear maps among them (without any thought to relations in general); I would prefer, instead, to start by specifying linearity as a property of general relations, then to illustrate the property with special cases that are mappings (linear maps) and collections (identity linear maps, construed as linear spaces). However, it seems I need to start by specifying an additive structure, whose domain is the linear space; and this amounts to the same thing as specifying the linear space at the outset.

There are spaces – of positions – which don't formally support an addition but do support a vector space of translations, differences or displacements whose action on position-space has enough of the character of an addition that if one (arbitrarily) chooses a position to treat as origin, one can identify each position with a translation – which maps the selected origin to the given position – and thereby induce an additive structure on positions. None the less, this additive structure is not intrinsic to the space – it is an artifact of our choice of origin – although the addition on our space of displacements is intrinsic and so is their action (in the guise of translations) on the members of the underlying space.

In general, an addition is a combiner and can be expressed as a mapping which, given one of the things the combiner knows how to add (to other things), maps that to a translation among the other things. In the case of a space of positions, any choice of origin provides an embedding of the space in its associated space of translations; in the case of a true addition, it so happens that there is a canonical one of these (which uses the additive identity as origin). We may thus unify the two cases by discussing the space of translations, anchored by: a requirement that there be some suitable embedding of the underlying space in its space of translations, and; a characterization of the action of translations and its relationship to such embeddings.

The principal problem of constructing only rational scalings is that a real vector space of modest dimension can be construed as a vector space over the rationals of infinitely larger dimension; however, if the space itself provides its own topology, I suppose we can specify scalings as those continuous automorphisms of the addition which commute with all others. Continuity of the additive structure ensures the positive rationals show up among the scalings; the topology of the space itself should then do topological completion for us, at least if we can manage to establish an ordering.

Once one has linear spaces, one can discuss simplices (which generalize 2-D triangles and 3-D tetrahedra to all dimensions); in a linear continuum, one is particularly interested in non-empty open simplices, since their edge-displacements span the space of translations (i.e. for a simplex with any interior, the edges out of any vertex form a basis). Suitably non-degenerate open simplices then open the door to bases; they also make it possible to specify chords of continuous (but not necessarily linear) maps from their neighbourhoods. Such chords necessarily imply linear maps we can construe as gradients, from which we can induce a specification of differentiability and of derivatives in linear continua. My hope is that this may be achieved without resort to any scalings but rationals, by exploiting the structures of the linear continua themselves; that is, topological completeness of the space of scalings of a linear space can flow from the algebra and topology of that space without having to be induced from (a topology on) the scalings we are obliged to construct in order to study the space.

Note that a given collection of values, even a given topology thereon, may support more than one binary operator which we can (albeit perhaps by violating our normal understanding of the values in question) interpret as addition and, from this, infer an additive continuum. This, in turn, provides for the possibility of a linear map from one additive structure to the other. On the space of linear maps from a linear space to itself, or indeed of scalings of the linear space, one has both the usual addition induced from point-wise addition at each input – u+v = (: u(x)+v(x) ←x :) – and their composition as mappings; if we construe the latter as an addition, the linear maps from the usual additive structure to this modified structure have the form of exponential functions. I'm fairly sure this is worth exploring.

The deep relationship between solving polynomial equations and selecting bases with respect to which assorted linear maps take canonical forms (diagonal for symmetric cases, a little more intricate for anti-symmetric) is also apt to be worthy of study.

See also

and 3blue1brown has a series on Essence of Linear Algebra which seems to do a tolerably good job of giving an intuitive sense for what it's all about, albeit relying on geometric concepts of length, area and volume that belong later in the theory (but really help with motivating why we build that theory).

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