Change of Basis - Preliminary Notions

Preliminary Notions

The standard basis for Rn is {e₁, ..., e_n}, where e_j is the element of Rn with 1 in the j-th place and 0s elsewhere.

If T : Rn → Rm is a linear transformation, the m × n matrix of T is the matrix t whose j-th column is T(e_j) for j = 1, ..., n. In this case we have T(x) = tx for all x in Rn, where we regard x as a column vector and the multiplication on the right side is matrix multiplication. It is a basic fact in linear algebra that the vector space Hom(Rn, Rm) of all linear transformations from Rn to Rm is naturally isomorphic to the space Rm × n of m × n matrices over R; that is, a linear transformation T : Rn → Rm is for all intents and purposes equivalent to its matrix t.

We will also make use of the following simple observation.

Theorem Let V and W be vector spaces, let {α₁, ..., α_n} be a basis for V, and let {γ₁, ..., γ_n} be any n vectors in W. Then there exists a unique linear transformation T : V → W with T(α_j) = γ_j for j = 1, ..., n.

This unique T is defined by T(x₁α₁ + ... + x_nα_n) = x₁γ₁ + ... + x_nγ_n. Of course, if {γ₁, ..., γ_n} happens to be a basis for W, then T is bijective as well as linear; in other words, T is an isomorphism. If in this case we also have W = V, then T is said to be an automorphism.

Now let V be a vector space over R and suppose {α₁, ..., α_n} is a basis for V. By definition, if ξ is a vector in V then ξ = x₁α₁ + ... + x_nα_n for a unique choice of scalars x₁, ..., x_n in R called the coordinates of ξ relative to the ordered basis {α₁, ..., α_n}. The vector x = (x₁, ..., x_n) in Rn is called the coordinate tuple of ξ (relative to this basis). The unique linear map φ : Rn → V with φ(e_j) = α_j for j = 1, ..., n is called the coordinate isomorphism for V and the basis {α₁, ..., α_n}. Thus φ(x) = ξ if and only if ξ = x₁α₁ + ... + x_nα_n.