Change of basis
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In linear algebra, a basis for a vector space of dimension n is a set of n vectors (α_{1}, …, α_{n}), called basis vectors, with the property that every vector in the space can be expressed as a unique linear combination of the basis vectors. The matrix representations of operators are also determined by the chosen basis. Since it is often desirable to work with more than one basis for a vector space, it is of fundamental importance in linear algebra to be able to easily transform coordinatewise representations of vectors and operators taken with respect to one basis to their equivalent representations with respect to another basis. Such a transformation is called a change of basis.
Although the terminology of vector spaces is used below and the symbol R can be taken to mean the field of real numbers, the results discussed hold whenever R is a commutative ring and vector space is everywhere replaced with free Rmodule.
Contents
Preliminary notions
Transformation matrix
The standard basis for is the ordered sequence , where is the element of with in the place and s elsewhere. For example, the standard basis for would be
If is a linear transformation, the matrix associated with is the matrix whose column is , for , that is
In this case we have , , where we regard 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() of all linear transformations from to is naturally isomorphic to the space of matrices over ; that is, a linear transformation is for all intents and purposes equivalent to its matrix .
Uniqueness of linear transformations
We will also make use of the following simple observation.
Theorem
Let and be vector spaces, let be a basis for , and let be any vectors in . Then there exists a unique linear transformation with , for .
This unique is defined by
Of course, if happens to be a basis for , then is bijective as well as linear; in other words, is an isomorphism. If in this case we also have , then is said to be an automorphism.
Coordinate isomorphism
Now let be a vector space over and suppose is a basis for . By definition, if is a vector in , then for a unique choice of scalars called the coordinates of relative to the ordered basis . The vector is called the coordinate tuple of relative to .
The unique linear map with for is called the coordinate isomorphism for and the basis . Thus if and only if .
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 in the vector space change when we select another basis.
Two dimensions
This means that given a matrix whose columns are the vectors of the new basis of the space (new basis matrix), the new coordinates for a column vector are given by the matrix product . 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 leftmultiply its components by the inverse of this matrix.^{[1]}
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 leftmultiplying its components by the inverse of this matrix.
General case
Suppose and are two ordered bases for an ndimensional vector space V over a field K. Let φ_{A} and φ_{B} be the corresponding coordinate isomorphisms (linear maps) from K^{n} to V, i.e. and for i = 1, …, n, where e_{i} denotes the ntuple with i^{ th} entry equal to 1, and all other entries equal to 0.
If is the coordinate ntuple of a vector v in V with respect to the basis A, so that , then the coordinate tuple of v with respect to B is the tuple y such that , i.e. , so that for any vector in V, the map maps its coordinate tuple with respect to A to its coordinate tuple with respect to B. Since this map is an automorphism on K^{n}, it therefore has an associated square matrix C. Moreover, the i^{ th} column of C is , that is, the coordinate tuple of α_{i} with respect to B.
Thus, for any vector v in V, if x is the coordinate tuple of v with respect to A, then the tuple is the coordinate tuple of v with respect to B. The matrix C is called the transition matrix from A to B.
The matrix of a linear transformation
Now suppose T : V → W is a linear transformation, {α_{1}, …, α_{n}} is a basis for V and {β_{1}, …, 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 T_{1} = ψ^{−1} ∘ T ∘ φ is a linear transformation from R^{n} 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 {α_{1}, …, α_{n}} and {β_{1}, …, 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 {α_{1}, …, α_{n}}, and we set y = tx and η = ψ(y), then η = ψ(T_{1}(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 {α_{1}, …, α_{n}} and {β_{1}, …, 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 φ_{1} and φ_{2} be the coordinate isomorphisms taking the usual basis in R^{n} to the first and second bases for V, and let ψ_{1} and ψ_{2} be the isomorphisms taking the usual basis in R^{m} to the first and second bases for W.
Let T_{1} = ψ_{1}^{−1} ∘ T ∘ φ_{1}, and T_{2} = ψ_{2}^{−1} ∘ T ∘ φ_{2} (both maps taking R^{n} to R^{m}), and let t_{1} and t_{2} be their respective matrices. Let p and q be the matrices of the changeofcoordinates automorphisms φ_{2}^{−1} ∘ φ_{1} on R^{n} and ψ_{2}^{−1} ∘ ψ_{1} on R^{m}.
The relationships of these various maps to one another are illustrated in the following commutative diagram. Since we have T_{2} = ψ_{2}^{−1} ∘ T ∘ φ_{2} = (ψ_{2}^{−1} ∘ ψ_{1}) ∘ T_{1} ∘ (φ_{1}^{−1} ∘ φ_{2}), and since composition of linear maps corresponds to matrix multiplication, it follows that
 t_{2} = q t_{1} p^{−1}.
Given that the change of basis has once the basis matrix and once its inverse, these objects are said to be 1co, 1contravariant.
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 {β_{1}, …, β_{n}} = {α_{1}, …, 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
 t_{2} = p t_{1} p^{−1}.
In this situation the invertible matrix p is called a changeofbasis matrix for the vector space V, and the equation above says that the matrices t_{1} and t_{2} 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 is defined by
If and 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 to 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:
 In the splitcomplex number plane there is an alternative "diagonal basis". The standard hyperbola xx − yy = 1 becomes xy = 1 after the change of basis. Transformations of the plane that leave the hyperbolae in place correspond to each other, modulo a change of basis. The contextual difference is profound enough to then separate Lorentz boost from squeeze mapping. A panoramic view of the literature of these mappings can be taken using the underlying change of basis.
 With the 2 × 2 real matrices one finds the beginning of a catalogue of linear algebras due to Arthur Cayley. His associate James Cockle put forward in 1849 his algebra of coquaternions or splitquaternions, which are the same algebra as the 2 × 2 real matrices, just laid out on a different matrix basis. Once again it is the concept of change of basis that synthesizes Cayley’s matrix algebra and Cockle’s coquaternions.
 A change of basis turns a 2 × 2 complex matrix into a biquaternion.
See also
 Coordinate vector
 Integral transform, the continuous analogue of change of basis.
 Active and passive transformation
References
 ^ "Change of Basis  HMC Calculus Tutorial". www.math.hmc.edu. Retrieved 20170822. and the explanation / proof "Why?". www.math.hmc.edu. Retrieved 20170822.
External links
 MIT Linear Algebra Lecture on Change of Basis, from MIT OpenCourseWare
 Khan Academy Lecture on Change of Basis, from Khan Academy