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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]].

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.

== Examples ==

== Examples ==