Website owner:  James Miller

         LINEAR SPACE (ABSTRACT VECTOR SPACE)

The definition of an abstract vector space is an obscure, abstruse, axiomatic one as are the definitions of many concepts of modern, abstract, axiom-based mathematics. A set of abstract algebraic laws or properties are stated as postulates that must be satisfied in order for the system to qualify as a vector space. These postulates derive from a list of properties of n-dimensional space.

Def. Linear Space (or Abstract Vector Space). A linear space is a set V of elements, called vectors, of an arbitrary nature for which the concepts of a sum and scalar multiplication (i.e. a product by a number) make sense, which is closed with respect to addition and scalar multiplication, and which satisfy the following postulates for any element x,y and z in V and any scalars a,b and c in a number field F:

  1. (x+y) + z = x + (y+z)

  2. There exists a "null" element 0 such that x+0 = x for every x in V

  3. For every element x there exists an opposite -x such that x+(-x) = 0

  4. x + y = y + x

  5. 1x = x for the unit 1 in F and for all x

  6. a(bx) = (ab)x

  7. (a+b)x = ax + bx

  8. c(x+y) = cx + cy

Postulates 1 through 4 are equivalent to the statement that V is an Abelian group with addition as the group operation.