FOM: Independence of "linear transformation" and "vector space"
Vaughan R. Pratt
pratt at cs.Stanford.EDU
Sun Nov 23 13:06:32 EST 1997
The question of whether the notion of vector space is prior to that of
linear transformation would seem to be more a matter of semantics than
of mathematics.
Consider the notion of "distance between vertices of a graph embedded
in the plane." Is "vertex of a graph" a prior notion? It certainly
isn't for the notion of distance in the plane, and it is surely a
semantical quibble whether the specialization of the notion of distance
to graph vertices makes "graph vertex" a prior notion.
The same holds for linear transformations, which are simply
homomorphisms of algebraic structures, specialized to the case where
those structures are vector spaces. Both the structures and their
homomorphisms exist a priori, it only remains to single out the vector
spaces.
As a practical matter it is not unreasonable to name the homomorphisms
before completing the identification of the class. This happens in the
following gedanken textbook treatment of vector spaces, whose order of
doing business is justified by the commonsense rule of "getting the
easy stuff out of the way first."
"A *vector space over* a field K is an algebra (A,+,<s_k>) where
+:A^2->A is a binary operation and <s_k> is a K-indexed family of unary
operations s_k:A->A, satisfying the equations given below. Vector
spaces over K and their homomorphisms, called *linear transformations*,
form a category denoted Vct_K. The equations are as follows."
The definition of "vector space" is not complete until the equations
have been given, yet "linear transformation" has already been defined.
Traditionally one completes the definition of the objects before naming
the associated homomorphisms. However this is more a cultural issue
than a logical necessity. One can easily imagine a curriculum that
first treated a few representative algebraic structures---monoids,
groups, lattices, Boolean algebras---then with those as the motivating
examples treated the general concepts of algebra, homomorphism,
subalgebra, and direct product, then leveraged those concepts to make
light work of a long list of further algebraic structures. With such a
curriculum one might well prefer to name the homomorphisms at the same
time the objects are named, instead of putting it off until after the
objects have been fully defined.
Vaughan Pratt
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