# vector space

## vector space

noun Mathematics.
an additive group in which addition is commutative and with which is associated a field of scalars, as the field of real numbers, such that the product of a scalar and an element of the group or a vector is defined, the product of two scalars times a vector is associative, one times a vector is the vector, and two distributive laws hold.
Also called

Origin:
1940–45

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World English Dictionary
 vector space —n maths See also scalar multiplication a mathematical structure consisting of a set of objects (vectors) associated with a field of objects (scalars), such that the set constitutes an Abelian group and a further operation, scalar multiplication, is defined in which the product of a scalar and a vector is a vector

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American Heritage
Science Dictionary
 vector space   A set of generalized vectors and a field of scalars, together with rules for their addition and multiplication (the same rules used for ordinary vectors and scalars).
The American Heritage® Science Dictionary
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FOLDOC
Computing Dictionary

### vector space definition

mathematics
An additive group on which some (scalar) field has an associative multiplicative action which distributes over the addition of the vector space and respects the addition of the (scalar) field: for vectors u, v and scalars h, k; h(u+v) = hu + hv; (h+k)u = hu + ku; (hk)u = h(ku).
[Simple example?]
(1996-09-30)

The Free On-line Dictionary of Computing, © Denis Howe 2010 http://foldoc.org
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Example sentences
The addition and the scalar multiplication induce on a real vector space structure.
Most of these rely on complicated geometrical constructions or theorems drawn from vector space theory.
The secret of quantum error correction is to encode a quantum state in a cleverly selected subspace of a larger vector space.
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