Changes

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—see below.  For the straightforward case, think of a scalar as just an ordinary [[real number]].)

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—see below.  For the straightforward case, think of a scalar as just an ordinary [[real number]].)

Not everything is a vector: some examples of things that are not vectors are air temperature and pressure, or the electrostatic voltage.  These have no direction.  They are scalars.  But there is a special type of [[derivative]], the "[[gradient]]" of a scalar, that is a vector.  This measures the ''change'' in a scalar from one point in space to another.

Not everything is a vector: some examples of things that are not vectors are air temperature and pressure, or the electrostatic voltage.  These have no direction.  They are scalars.  But there is a special type of [[Derivative (calculus)|derivative]], the "[[gradient]]" of a scalar, that is a vector.  This measures the ''change'' in a scalar from one point in space to another.

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

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