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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 v1, v2, ... , vn such that every member v of V can be written as a linear combination

v = a1v1 + a2v2 + .... + anvn

(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 v1, v2, ... , vn are said to form a basis for V.

If v1, v2, ... , vn form a basis for a vector space V over a field F, then for each v of V the representation

v = a1v1 + a2v2 + .... + anvn

of v as a linear combination over F of v1, v2, ... , vn is unique. The numbers a1, a2, ... , an are called the coordinates of vector v with respect to the basis { v1, v2, ... , vn } of vector space V. The n-tuple (a1, a2, ... , an ) is called the coordinate vector of v relative to the basis {v1, v2, ... , vn } .

Example. Consider the set of all polynomials

a0 + a1x + a2x2 + ... + anxn

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 + ... + anxn

with coordinates a1, a2, ... , an relative to the basis {1, x, x2, ... , xn }.

References.

Hohn, Elem, Matrix Theory. p.87

Lipchutz. Linear Algebra. p. 92

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