Website owner:  James Miller

     COORDINATES OF A VECTOR IN AN ABSTRACT VECTOR SPACE 

An abstract vector space V is called finite dimensional of dimension n if there exist n members v_1,v₂, ... ,v_n such that every member v of V can be written as a linear combination

               v = a₁v₁ + a₂v₂ + .... + a_nv_n

(where the a’s belong to some number field F) and there exist no set of fewer than n members of V with the same property. The vectors v_1,v₂, ... ,v_n are said to form a basis for V.

If v_1,v₂, ... ,v_n form a basis for a vector space V over a field F, then for each v of V the representation

              v = a₁v₁ + a₂v₂ + .... + a_nv_n

of v as a linear combination over F of v_1,v₂, ... ,v_n is unique. The numbers a_1,a₂, ... ,a_n are called the coordinates of vector v with respect to the basis { v_1,v₂, ... ,v_n } of vector space V. The n-tuple (a_1,a₂, ... ,a_n ) is called the coordinate vector of v relative to the basis { v_1,v₂, ... ,v_n } .

Example. Consider the set of all polynomials

                         a₀ + a₁x + a₂x² + ... + a_nxⁿ

of degree n. These polynomials can be viewed as vectors in a finite dimensional vector space V of dimension n+1 since they can be represented as linear combinations of the set of vectors

                          1,x,x², ... ,xⁿ

and the vectors 1,x,x², ... ,xⁿ are linearly independent and their number is n+1. The set {1,x,x², ... ,xⁿ } can be taken as a basis for this space V and any vector in V corresponds to a polynomial

                a₀ + a₁x + a₂x² + ... + a_nxⁿ

with coordinates a_1,a₂, ... ,a_n relative to the basis {1,x,x², ... ,xⁿ }.

References.

  Hohn, Elem, Matrix Theory. p.87

  Lipchutz. Linear Algebra. p. 92