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A '''vector space''' is a construct in mathematics that generalizes the familiar notion of vectors in the x-y plane.  Recall that given two vectors <math>\vec{v} = (v_1,v_2)</math> and <math>\vec{w} = (w_1,w_2)</math>, we can take form the sum <math>\vec{v}+\vec{w}=(v_1+v_2,w_1+w_2)</math> of two vectors, and the product of a vector with a scalar (i.e., a [[real number]]), by setting <math> a \cdot \vec{v} = (av_1,av_2)</math>.  A vector space consists of a collection of objects that has two analogous operations: it is possible to add two of the objects together, and it is possible to multiply one by a scalar.  Vector spaces are the fundamental objects of study of [[linear algebra]] and, due to their usefulness with [[gradient]]s, advanced calculus.

A '''vector space''' is one of the most fundamental and important concepts in mathematics, physics, and engineering.  It has its origins in the notion of a "direction" and a "magnitude".  Perhaps the simplest vector to visualize is the ''velocity vector'', showing the speed and direction of motion of a particle.  Other extremely common vectors are the [[electric field]] and [[magnetic field]] vectors, though vectors abound in numerous areas of mathematics and physics.

 

 

A vector space is a set of vectors that can be added to each other or multiplied by a scalar.  (The term "scalar" is used for treatments of unusual vector spaces&mdash;see below.  For the straightforward case, think of a scalar as just an ordinary [[real number]].)

 

Vector spaces have a "dimension".  In the physically simple cases, that dimension is usually just 2 or 3.  Vectors drawn as arrows on a piece of paper are two-dimensional vectors.  Vectors giving velocity, electric field strength, etc., in real 3-dimensional space are three-dimensional vectors.  Given a choice of "coordinate system" or "basis" for representing vectors, any vector can be denoted by 2 or 3 (or whatever the dimension is) scalars.  So, for example, a particle's velocity vector can be represented by its x-velocity, y-velocity, and z-velocity.  These numbers are called the "components" of the vector, and are generally written with subscripts running from 1 to the dimension of the space.  So a vector might be represented as

 

:<math>\vec{v} = (v_1,v_2,v_3)</math>

and

:<math>\vec{w} = (w_1,w_2,w_3)</math>

 

When represented in this way, the vector sum is very straightforward:

 

:<math>\vec{v}+\vec{w}=(v_1+v_2+v_3,w_1+w_2+w_3)</math>

 

and the vector-scalar product is equally straightforward:

 

:<math> a \cdot \vec{v} = (a v_1,a v_2,a v_3)</math>

 

Vector spaces are the fundamental objects of study of [[linear algebra]] and, due to their usefulness with [[gradient]]s, advanced calculus.

== Examples ==

== Examples ==

The above discussion only considers vector spaces in which the scalars are real numbers, but we could just as well talk about the set of polynomials with complex coefficients, where we multiple by complex scalars.  More generally, given any [[field]] <math>k</math>, a vector space over <math>k</math> is 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.  In terms of another definition, a vector space is simply a module for which the ground ring is a field.

The above discussion only considers vector spaces in which the scalars are real numbers, but we could just as well talk about the set of polynomials with complex coefficients, where we multiple by complex scalars.  More generally, given any [[field]] <math>k</math>, a vector space over <math>k</math> is 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.  In terms of another definition, a vector space is simply a module for which the ground ring is a field.

[[Category:Algebra]]

[[Category:Algebra]]