Change of basis


In linear algebra, a basis for a vector space is a linearly independent set spanning the vector space. This article deals mainly with finite-dimensional vector spaces, but many of the theorems are also valid for infinite-dimensional vector spaces. A basis for a vector space of dimension n is a set of n vectors, 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 coordinate-wise 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 symbol R used below can be taken to mean the field of real numbers, the results are valid if R is replaced by any field F. Although the terminology of vector spaces is used below, the results discussed hold whenever R is a commutative ring and vector space is everywhere replaced with free R-module.

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 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.

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, 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 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 :
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 and are two ordered bases for an n-dimensional vector space V over a field K. Let φA and φB be the corresponding coordinate isomorphisms from Kn to V, i.e. and for, where ei denotes the n-tuple with i th entry equal to 1, and all other entries equal to 0.
If is the coordinate n-tuple 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 Kn, it therefore has an associated square matrix C. Moreover, the i th column of C is, that is, the coordinate tuple of αi 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 is a linear transformation, is a basis for V and 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 is a linear transformation from Rn to Rm, and therefore has a matrix t; its jth column is for. This matrix is called the matrix of T with respect to the ordered bases and If and y and x are the coordinate tuples of η and ξ, then. Conversely, if ξ is in V and is the coordinate tuple of ξ with respect to and we set and, then. That is, if ξ is in V and η is in W and x and y are their coordinate tuples, then if and only if.
Theorem Suppose U, V and W are vector spaces of finite dimension and an ordered basis is chosen for each. If and are linear transformations with matrices s and t, then the matrix of the linear transformation is st.

Change of basis

Now we ask what happens to the matrix of when we change bases in V and W. Let and be ordered bases for V and W respectively, and suppose we are given a second pair of bases and 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 Rm to the first and second bases for W.
Let, and , and let t1 and t2 be their respective matrices. Let p and q be the matrices of the change-of-coordinates automorphisms on Rn and on Rm.
The relationships of these various maps to one another are illustrated in the following commutative diagram.
Since we have, 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, these 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.
We can naturally take and The matrix of the linear map T is necessarily square.

Change of basis

We apply the same change of basis, so that 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 which is linear in both arguments. That is, 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: