Change of basis

Transformation matrix

The standard basis for R n is the ordered sequence E n = { e 1 , ⋯ , e n } , where e j is the element of R n with 1 in the j t h place and 0 s elsewhere. For example, the standard basis for R 2 would be

If T : R n → R m is a linear transformation, the m × n matrix associated with T is the matrix M T whose j t h column is T ( e j ) ∈ R m , for j = 1 , ⋯ , n , that is

In this case we have T ( x ) = M T ⋅ x , ∀ x ∈ R n , 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 H o m ( R n , R m ) of all linear transformations from R n to R m is naturally isomorphic to the space R m × n of m × n matrices over R ; that is, a linear transformation T : R n → R m is for all intents and purposes equivalent to its matrix M T .

Uniqueness linear transformation

We will also make use of the following simple observation.

Theorem

Let V and W be vector spaces, let B = { α 1 , ⋯ , α n } be a basis for V , and let C = { γ 1 , ⋯ , 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

Of course, if C = { γ 1 , ⋯ , 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.

Coordinate isomorphism

Now let V be a vector space over R and suppose B = { α 1 , ⋯ , α n } is a basis for V . By definition, if ξ is a vector in V , then ξ = x 1 α 1 + ⋯ + x n α n for a unique choice of scalars x 1 , ⋯ , x n ∈ R called the coordinates of ξ relative to the ordered basis B . The vector x = ( x 1 , ⋯ , x n ) T ∈ R n is called the coordinate tuple of ξ relative to B .

The unique linear map ϕ : R n → V with ϕ ( e j ) = α j for j = 1 , ⋯ , n is called the coordinate isomorphism for V and the basis B = { α 1 , ⋯ , α n } . Thus ϕ ( x ) = ξ if and only if ξ = x 1 α 1 + ⋯ + x n α n .

Matrix of a set of vectors

A set of vectors can be represented by a matrix of which each column consists of the components of the corresponding vector of the set. As a basis is a set of vectors, a basis can be given by a matrix of this kind. Later it will be shown that the change of basis of any object of the space is related to this matrix. For example vectors change with its inverse (and they are therefore called contravariant objects).

Change of coordinates of a vector

First we examine the question of how the coordinates of a vector ξ in the vector space V , change when we select another basis.

Two dimensions

This means that given a matrix M whose columns are the vectors of the new basis of the space (new basis matrix), the new coordinates for a column vector v are given by the matrix product M − 1 v . For this reason, it is said that ordinary vectors are contravariant objects.

Any finite set of vectors can be represented by a matrix in which its columns are the coordinates of the given vectors. As an example in dimension 2, a pair of vectors obtained by rotating the standard basis counterclockwise for 45°. The matrix whose columns are the coordinates of these vectors is

If we want to change any vector of the space to this new basis, we only need to left-multiply its components by the inverse of this matrix.

Three dimensions

For example, let R be a new basis given by its Euler angles. The matrix of the basis will have as columns the components of each vector. Therefore, this matrix will be (See Euler angles article):

Again, any vector of the space can be changed to this new basis by left-multiplying its components by the inverse of this matrix.

General case

Suppose A = { α 1 , ⋯ , α n } and B = { β 1 , ⋯ , β n } are two ordered bases for V . Let φ₁ and φ₂ be the corresponding coordinate isomorphisms (linear maps) from Rⁿ to V, i.e. φ₁(e_j) = α_j and φ₂(e_j) = α′_j for j = 1, …, n.

If x = ( x 1 , ⋯ , x n ) is the coordinate n -tuple of ξ with respect to the first basis, so that ξ = ϕ 1 ( x ) , then the coordinate tuple of ξ with respect to the second basis is ϕ 2 − 1 ( ξ ) = ϕ 2 − 1 ( ϕ 1 ( x ) ) . Now the map ϕ 2 − 1 ∘ ϕ 1 is an automorphism on R n and therefore has a matrix C . Moreover, the j t h column of C is ϕ 2 − 1 ∘ ϕ 1 ( e j ) = ϕ 2 − 1 ( α j ) , that is, the coordinate n -tuple of α j with respect to the second basis B = { β 1 , ⋯ , β n } . Thus y = ϕ 2 − 1 ( ϕ 1 ( x ) ) = C x is the coordinate n -tuple of ξ with respect to the basis B = { β 1 , ⋯ , β n }

The matrix of a linear transformation

Now suppose T : V → W is a linear transformation, {α₁, …, α_n} is a basis for V and {β₁, …, β_m} is a basis for W. Let φ and ψ be the coordinate isomorphisms for V and W, respectively, relative to the given bases. Then the map T₁ = ψ⁻¹ ∘ T ∘ φ is a linear transformation from Rⁿ to R^m, and therefore has a matrix t; its jth column is ψ⁻¹(T(α_j)) for j = 1, …, n. This matrix is called the matrix of T with respect to the ordered bases {α₁, …, α_n} and {β₁, …, β_m}. If η = T(ξ) and y and x are the coordinate tuples of η and ξ, then y = ψ⁻¹(T(φ(x))) = tx. Conversely, if ξ is in V and x = φ⁻¹(ξ) is the coordinate tuple of ξ with respect to {α₁, …, α_n}, and we set y = tx and η = ψ(y), then η = ψ(T₁(x)) = T(ξ). That is, if ξ is in V and η is in W and x and y are their coordinate tuples, then y = tx if and only if η = T(ξ).

Theorem Suppose U, V and W are vector spaces of finite dimension and an ordered basis is chosen for each. If T : U → V and S : V → W are linear transformations with matrices s and t, then the matrix of the linear transformation S ∘ T : U → W (with respect to the given bases) is st.

Change of basis

Now we ask what happens to the matrix of T : V → W when we change bases in V and W. Let {α₁, …, α_n} and {β₁, …, β_m} be ordered bases for V and W respectively, and suppose we are given a second pair of bases {α′₁, …, α′_n} and {β′₁, …, β′_m}. Let φ₁ and φ₂ be the coordinate isomorphisms taking the usual basis in Rⁿ to the first and second bases for V, and let ψ₁ and ψ₂ be the isomorphisms taking the usual basis in R^m to the first and second bases for W.

Let T₁ = ψ₁⁻¹ ∘ T ∘ φ₁, and T₂ = ψ₂⁻¹ ∘ T ∘ φ₂ (both maps taking Rⁿ to R^m), and let t₁ and t₂ be their respective matrices. Let p and q be the matrices of the change-of-coordinates automorphisms φ₂⁻¹ ∘ φ₁ on Rⁿ and ψ₂⁻¹ ∘ ψ₁ on R^m.

The relationships of these various maps to one another are illustrated in the following commutative diagram.

Since we have T₂ = ψ₂⁻¹ ∘ T ∘ φ₂ = (ψ₂⁻¹ ∘ ψ₁) ∘ T₁ ∘ (φ₁⁻¹ ∘ φ₂), and since composition of linear maps corresponds to matrix multiplication, it follows that

Given that the change of basis has once the basis matrix and once its inverse, this objects are said to be 1-co, 1-contra-variant.

The matrix of an endomorphism

An important case of the matrix of a linear transformation is that of an endomorphism, that is, a linear map from a vector space V to itself: that is, the case that W = V. We can naturally take {β₁, …, β_n} = {α₁, …, α_n} and {β′₁, …, β′_m} = {α′₁, …, α′_n}. The matrix of the linear map T is necessarily square.

Change of basis

We apply the same change of basis, so that q = p and the change of basis formula becomes

In this situation the invertible matrix p is called a change-of-basis matrix for the vector space V, and the equation above says that the matrices t₁ and t₂ are similar.

The matrix of a bilinear form

A bilinear form on a vector space V over a field R is a mapping V × V → R which is linear in both arguments. That is, B : V × V → R is bilinear if the maps

are linear for each w in V. This definition applies equally well to modules over a commutative ring with linear maps being module homomorphisms.

The Gram matrix G attached to a basis α 1 , … , α n is defined by

If v = ∑ i x i α i and w = ∑ i y i α i are the expressions of vectors v, w with respect to this basis, then the bilinear form is given by

The matrix will be symmetric if the bilinear form B is a symmetric bilinear form.

Change of basis

If P is the invertible matrix representing a change of basis from α 1 , … , α n to α 1 ′ , … , α n ′ then the Gram matrix transforms by the matrix congruence

Important instances

In abstract vector space theory the change of basis concept is innocuous; it seems to add little to science. Yet there are cases in associative algebras where a change of basis is sufficient to turn a caterpillar into a butterfly, figuratively speaking: