Basis of a Vector Space
Overview
A basis of a vector space is a set of vectors for which any other vector in the vector space can be
written as a sum over.
In the finite case, there is a set of linearly independent (see below) vectors
{% v_1,v_2,...,v_n %}
such that, for any vector {% \vec{v} %} in the vector space, {% \vec{v} %} can be expressed as a sum
{% \vec{v} = a_1v_1 + a_2 v_2 +... + a_n v_n %}
When any vector in a vector space can be expressed this way, the vector space is known as a finite dimensional vector space.
(finite in the sense that the basis set is finite)
Dependent and Independent Vectors
A set of vectors {% v_1,v_2,...,v_n %} is called
linearly independent if
the only solution to the equation
{% a_1 v_1 + a_2 v_2 + ... + a_n v_n = 0 %}
is {% a_1=0, a_2 =0 ,... , a_n =0 %}
If the set of vectors is not linearly independent, then they are known as dependent.
Definitions and Theorems
- Theorem - Every basis of a finite dimensional vector space has the same number of elements. THis number
is referred to as the Dimension of the vector space
- Existence
- details the theorems concerning existence of a basis
