How do you construct an ordered basis No idea how 100 articl

How do you construct an ordered basis? No idea how, 100+ articles on the internet and not one of them explains what an ordered basis is or how to find it.

Let\'s say I have a matrix {{a,b},{c,d}} and another matrix {{e,f,g},{h,i,j}}...what are the ordered basis for these matrices?

Solution

A set of vectors in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.

It is often convenient to list the basis vectors in a specific order, for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span.

Ordered bases and coordinates

A basis is just a linearly independent set of vectors with or without a 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 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

: Fⁿ V

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

v_i = (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

(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 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 determined by an ordered basis {v_i} equips V with coordinates: if, for a vector v V, 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 ¹ is linear. Hence they are linear functionals. They form a basis for thedual space of V, called the dual basis.