Abstract Vector Space

An abstract vector space of dimension n over a field k is the set of all formal expressions

where {v_1,v_2,...,v_n} is a given set of n objects (called a basis) and (a_1,a_2,...,a_n) is any n-tuple of elements of k. Two such expressions can be added together by summing their coefficients,

This addition is a commutative group operation, since the zero element is 0v_1+0v_2+...+0v_n and the inverse of a_1v_1+a_2v_2+...+a_nv_n is (-a_1)v_1+(-a_2)v_2+...+(-a_n)v_n. Moreover, there is a natural way to define the product of any element a_1v_1+a_2v_2+...+a_nv_n by an arbitrary element (a so-called scalar) a of k,

Note that multiplication by 1 leaves the element unchanged.

This structure is a formal generalization of the usual vector space over R^n, for which the field of scalars is the real field R and a basis is given by {(1,0,0,...,0),(0,1,0,0,...,0),...,(0,0,...,0,1)}. As in this special case, in any abstract vector space V, the multiplication by scalars fulfils the following two distributive laws:

1. For all a,b in k and all v in V, (a+b)v=av+bv.

2. For all a in k and all v,w in V, a(v+w)=av+aw.

These are the basic properties of the integer multiples in any commutative additive group. This special behavior of a product with respect to the sum defines the notion of linear structure, which was first formulated by Peano in 1888.

Linearity implies, in particular, that the zero elements 0_k and 0_V of k and V annihilate any product. From (1), it follows that

for all v in V, whereas from (2), it follows that

for all a in k.

A more general kind of abstract vector space is obtained if one admits that the basis has infinitely many elements. In this case, the vector space is called infinite-dimensional and its elements are the formal expressions in which all but a finite number of coefficients are equal to zero.

See also

Free Module, Quotient Vector Space, Vector Space

This entry contributed by Margherita Barile

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References

Peano, G. Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann. Torino, Italia: Fratelli Bocca, 1888.

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Abstract Vector Space

Cite this as:

Barile, Margherita. "Abstract Vector Space." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/AbstractVectorSpace.html

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