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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
v_{1,} v_{2}, ... , v_{n} such that every member v of V can be written as a linear combination

v = a_{1}v_{1} + a_{2}v_{2} + .... + a_{n}v_{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,} v_{2}, ... , v_{n} are said to form a basis for V.

If v_{1,} v_{2}, ... , v_{n} form a basis for a vector space V over a field F, then for each v of V the
representation

v = a_{1}v_{1} + a_{2}v_{2} + .... + a_{n}v_{n}

of v as a linear combination over F of v_{1,} v_{2}, ... , v_{n} is unique. The numbers a_{1,} a_{2}, ... , a_{n} are
called the *coordinates* of vector v with respect to the basis { v_{1, }v_{2}, ... , v_{n} } of vector space V.
The n-tuple (a_{1,} a_{2}, ... , a_{n} ) is called the *coordinate vector *of v relative to the basis {v_{1,} v_{2}, ... , v_{n} }
.

Example. Consider the set of all polynomials

a_{0} + a_{1}x + a_{2}x^{2} + ... + a_{n}x^{n}

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, x^{2}, ... , x^{n}

and the vectors 1, x, x^{2}, ... , x^{n} are linearly independent and their number is n+1. The set {1, x,
x^{2}, ... , x^{n} } can be taken as a basis for this space V and any vector in V corresponds to a
polynomial

a_{0} + a_{1}x + a_{2}x^{2} + ... + a_{n}x^{n}

with coordinates a_{1, }a_{2}, ... , a_{n} relative to the basis {1, x, x^{2}, ... , x^{n} }.

References.

Hohn, Elem, Matrix Theory. p.87

Lipchutz. Linear Algebra. p. 92

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