Overview
Vector spaces are ubiquitous in mathematics and physics. They are defined to be a collection of objects equipped with an ability to add any two objects, and to multiply any object by a scalar. Examples include the real numbers, complex numbers, column vectors and matrices.
Axioms
A vector space is a set (whose members are called vectors) that comes equipped with two operations.
A notion of addition of two vectors that returns a third vector.
{% \vec{V}_1 + \vec{V}_2 = \vec{V}_3 %}
And a notion of multiplication with members of a
given
Field,
that also returns another member of the set of vectors.
{% a\vec{V_1} = \vec{V}_2 %}
In addition, the following must hold.
- Closure under Addition
For every {% x %} and {% y %} in {% V %}, there is a unique element in {% V %}, {% x+y %}
- Associativity of Addition
{% (x+y)+z = x+(y+z) %}
- Closure under Scalar Multiplication
For every element {% x %} in {% V %}, and number {% a %} in the field {% \mathbb{F} %}, there is a unique element in {% V %}, {% ax %}.
- Existence of Zero Vector
There exists a zero vector, {% 0 %}, such that {% x+0=0+x=x %}
- Existence of Inverse
For every vector {% x %}, there is another vector,{% -x %}, which added to {% x %} yields the zero vector.