Basis (linear Algebra) - Ordered Bases and Coordinates

Ordered Bases and Coordinates

A basis is just a set of vectors with no given ordering. For many purposes it is convenient to work with an ordered basis. For example, when working with a coordinate representation of a vector it is customary to speak of the "first" or "second" coordinate, which makes sense only if an ordering is specified for the basis. For finite-dimensional vector spaces one typically indexes a basis {v_i} by the first n integers. An ordered basis is also called a frame.

Suppose V is an n-dimensional vector space over a field F. A choice of an ordered basis for V is equivalent to a choice of a linear isomorphism φ from the coordinate space Fn to V.

Proof. The proof makes use of the fact that the standard basis of Fn is an ordered basis.

Suppose first that

φ : Fn → V

is a linear isomorphism. Define an ordered basis {v_i} for V by

v_i = φ(e_i) for 1 ≤ i ≤ n

where {e_i} is the standard basis for Fn.

Conversely, given an ordered basis, consider the map defined by

φ(x) = x₁v₁ + x₂v₂ + ... + x_nv_n,

where x = x₁e₁ + x₂e₂ + ... + x_ne_n is an element of Fn. It is not hard to check that φ is a linear isomorphism.

These two constructions are clearly inverse to each other. Thus ordered bases for V are in 1-1 correspondence with linear isomorphisms Fn → V.

The inverse of the linear isomorphism φ determined by an ordered basis {v_i} equips V with coordinates: if, for a vector v ∈ V, φ−1(v) = (a₁, a₂,...,a_n) ∈ Fn, then the components a_j = a_j(v) are the coordinates of v in the sense that v = a₁(v) v₁ + a₂(v) v₂ + ... + a_n(v) v_n.

The maps sending a vector v to the components a_j(v) are linear maps from V to F, because of φ−1 is linear. Hence they are linear functionals. They form a basis for the dual space of V, called the dual basis.