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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.  Not everything is a vector: an example of something that is not a vector is temperature, as it has no direction.  It is called a "[[scalar]]" rather than a vector.  But the ''change'' in temperature is a vector.

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.

The most important operations involving vectors are the ''vector sum'' and the ''vector-scalar product''.  As an example of the first, if we are in a train traveling with speed given by one vector, and we throw something inside the train with a velocity, relative to the train, of another vector, the velocity of the object relative to a fixed observer is the sum of those two vectors.  As an example of the second, if we double the current through an electromagnet, its magnetic field vector will be multiplied by the number 2.  That is, its direction will be unchanged and its magnitude will double.

The most important operations involving vectors are the ''vector sum'' and the ''vector-scalar product''.  As an example of the first, if we are in a train traveling with speed given by one vector, and we throw something inside the train with a velocity, relative to the train, of another vector, the velocity of the object relative to a fixed observer is the sum of those two vectors.  As an example of the second, if we double the current through an electromagnet, its magnetic field vector will be multiplied by the number 2.  That is, its direction will be unchanged and its magnitude will double.

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]].)

 

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