]> Linearity

Linearity: the natural harmony of addition and scaling

Linear algebra emerges when a context deals with values that can be added and (in one sense or another) scaled, yielding values of the same kind (thus likewise amenable to addition and scaling).

A minimal formalisation of addition suffices to induce a basic form of scaling: repeated addition always implies a representation of the positive naturals among automorphisms of the addition. A suitable embedding of the addition's operands in its automorphisms can serve as a multiplication on those operands, forming a ringlet. In any case, the automorphisms form a ringlet under their pointwise addition, using composition as multiplication; and the action of the automorphisms on the addition is distinctly multiplicative. Selecting a commutative sub-ringlet of the automorphisms to construe as scalings, typically represented as a homomorphic image of some ringlet of scalars, lets us form a module out of our addition. For ringlets with particularly nice properties, the module may be called a linear or vector space and its homomorphisms linear maps. I'll refer to values of the module as scalable values for the present.


Given a specific scalable value a, the mapping on scalable values that just adds a to its input, add(a) = (: b+a ←b :), is described as translation by a (or through a); and the mapping on scalars that maps each to its result of scaling a, (: x.a ←x :), is known as a's ray. The crucial properties of the addition and scaling are:

Addition and multiplication must be Abelian combiners.
Abelian (a.k.a. commutative) just means you can swap the order of operands; a+b = b+a whenever either is meaningful; and a.b = b.a likewise. A combiner is just a binary operator that's associative and closed. A binary operator combines values two at a time. Being closed just means that combining two operands gives an operand; if we know how to multiply or add a and b, then a +b or a.b are also values we know how to multiply or add (in an appropriate sense). Associativity means that (a.b).c = a.(b.c) whenever both are meaningful, likewise (a+b)+c = a+(b+c); consequently, we can meaningfully talk about a.b.c or a+b+c without needing parentheses to indicate the order in which the additions or multiplicaions are to be performed. Taken together, these imply that we can define sums (of several scalable values or of several scalars) and products (of several scalars and at most one scalable value) without reference to the order in which we add or multiply the values.
Scaling must distribute over addition
With a, b as scalars and x, y as scaleable values, we require (a+b).x = (a.x)+(b.x) and a.(x+y) = (a.x)+(a.y). In consequence of this, taken together with the dropping of parentheses made possible by associativity, it is usual to omit the parentheses around products, writing a.(x+y) = a.x +a.y and similar; when several values and scalings are combined with no interposed parentheses, all multiplication is done first, then the addition; alternatively, this may be expressed as multiplication binding more tightly than addition.
Addition and non-zero scaling must be cancellable
i.e. each translation is monic, as is each non-zero scaling and the ray of each non-zero scalable value. Whenever both a.e and c.e are meaningful, a.e = c.e must imply either a = c or e is an additive identity. Likewise, whenever a+e and c+e are meaningful, a+e = c+e must imply a = c. Cancellability makes it possible to augment the scalings with ratios and the scalable values and scalings with differences so as to make the scalable values into an additive group, the scalings also into an additive group and the non-zero scalings into a multiplicative group. More orthodox treatments take at least some of these augmentations for granted and, when they take all, refer to the scalable values as vectors.

The actual scalings in use may be restricted to whole numbers or they may form a continuum such as the real numbers; there may (as arises among the complex numbers) be some scalars whose squares are additive inverses for 1, or there may be none such (as for reals and naturals). For the sake of generality, I shall thus refer to the collection of scalars in use simply as {scalars}.


Context may deal with more than one collection of scalable values; within each such collection, all of the above holds; if the ringlets of scalings in use by some such collections are isomorphic, we can represent the scalings as multiplication by a common set of scalars; this gives structure to the relationships between the disparate collections of scalable values. Given one collection, U, whose members may be scaled by {scalars} and added to one another, and a second collection V, whose members may likewise be scaled and added to one another, I describe a relation (V:r:U) as linear – formally: {scalars}-linear or linear over {scalars} – precisely if:

Using the notation for mappings, these can be summarised by r(u+a.w) = r(u) +a.r(w) for every u, w in U and scalar a; a linear mapping is also known as a linear map. When a collection (construed as the identity mapping on its members) is linear, these conditions simply say that adding members or scaling a member always yields a member; I describe such a collection as a linear space; when, furthermore, it is an additive group (as may be achieved by the augmentation mentioned above) it is known as a vector space and its members as vectors. The collections of left and right values of a linear relation are necessarily always linear spaces; and {scalars} itself is, by specification, always a {scalars}-linear space. The collection of linear maps from a linear space U to {scalars} is also necessarily a linear space; it is known as the dual of U, dual(U) = {linear maps ({scalars}: |U)}.

Linear maps from a module to itself are linear automorphisms; as ever, these form a ringlet; and the module can be construed as a module over the centre of this ringlet without changing its behaviour as a module. We may start with a commutative ringlet embedded in the automorphisms of an addition, to make the addition a module over the ringlet, and later find the module's ringlet of linear automorphisms has a centre bigger than the image of the original ringlet. It then makes sense to treat members of the centre as scalings also, implicitly extending the original ringlet. Conversely, we may have an addition among whose automorphisms there is some particular ringlet that is of interest to us (e.g. its members correspond to physically realisable transformations of some system described by the values being added); we can then find all the automorphisms of the addition that commute with all of these and make our addition a module over the ringlet of those.

If our ringlet has a conjugation, we can introduce a partner of linearity, called antlinearity: a mapping between modules over the ringlet is conjugate-linear if it is an automorphism of the underlying additions while conjugating scaling. A conjugate-linear map from the module to its dual then provides a way of combining two of our scalable values to produce a number, as does a linear map from the module to its dual; these serve as a way to multiply scalable values, to give a number; when they have suitable symmetries, they are known as hermitian and quadratic forms. These, in turn, let us take the square of a scalable value; in the hermitian case (conjugate-linear into the dual) the resulting number can always be real even if the ringlet isn't all real; and, when every non-zero scalable value has a positive square, in this sense, we can use it to define a sense of distance.

Span and bases

Given a relation r whose left values lie in one module and right values lie another (possibly the same one) over the same ringlet of scalars, the span of r (over a given ringlet, or the R-span if R is that ringlet; typically R is implicit from context) is the relation obtained from it by imposing linearity; formally

i.e. whenever f and g are equal-length lists of r's right and left values, respectively, and r relates each g(i) to the matching f(i), we can select an arbitrary list, of the same length, of scalars, apply each scaling to the matching enties in f and g, sum the results and span(r) will relate the scaled-g sum to the scaled-f sum. (Note that this only involves finite sums; this leads to some complications if there are any infinite linearly independent mappings.) This can be characterised as the linear completion of r. When r is simply a collection, its span is just the set of values one can obtain by scaling and summing r's members.

Given a mapping to a linear space, we can take the span of its collection of outputs (i.e. left values); if no proper sub-relation of the mapping has, as the span of its collection of outputs, the same linear space then the mapping is described as linearly independent; this necessarily implies that the mapping is also monic. A linearly independent mapping is known as a basis of the span of its collection of outputs. When a mapping (:b|n) is linearly independent, with V = span(|b:), a mapping (dual(V): p |n) is described as a dual basis of b precisely if: for each i in n, p(i,b(i)) = 1 and, for each other j in n, j ≠ i, p(i,b(j)) = 0. In such a case, any v in V can be written as sum(: (p(i)·v).b(i) ←i :n) and the span of any mapping (:f|b) is (: sum(: (p(i)·v).f(b(i)) ←i :n) ←v |V).

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