Change of basis

In linear algebra, change of basis refers to the conversion of vectors and linear transformations between matrix representations which have different bases.

Expression of a basis

Main article: Basis (linear algebra)

Basis defined by Euler angles; the xyz (fixed) system is shown in blue, the XYZ (rotated) system is shown in red. The line of nodes, labeled N, is shown in green.

There are several ways to describe a basis for the space. Some are made ad-hoc for a specific dimension. For example, there are several ways to give a basis in dim 3, like Euler angles.

The general case is to give the components of the new basis vectors in its current basis. This is also the more general method because it can express any possible set of vectors. This matrix can be seen as three things:

Basis Matrix: Is a matrix that represents the basis, because its columns are the vectors of the basis.

Rotation operator: When orthonormal bases are used, any other orthonormal basis can be defined by a rotation matrix. The matrix of the rotation operator that rotates the vectors of the basis to the new one. The matrix of the rotation operator is exactly the same matrix of vectors because the rotation matrix multiplied by the identity matrix I has to be the basis matrix.

Change of basis matrix: This matrix will be used to change different objects of the space to the new basis. Therefore is called "change of basis" matrix, though some objects change with its inverse. It also represents the coefficients of any vector of the new basis as linear combinations of the current basis.

Change of basis for vectors

Any vector $\vec u$ in n-dimensional space can be uniquely expressed by a linear combination of basis vectors $\vec v_1, \vec v_2, \ldots, \vec v_n$.

$\vec u = u_1 \vec v_1+ \ldots + u_n \vec v_n = [\vec v_1, \ldots, \vec v_n]\begin{bmatrix} u_1 \\u_2 \\ \vdots \\ u_n\end{bmatrix}$

If we are given a new basis W, we need to know the new vectors as linear span of the old ones:

$\vec w_i = w_{1i} \vec v_1 + w_{2i} \vec v_2 + \cdots + w_{ni} \vec v_n \qquad (1)$

It is customary to express the whole new basis coefficients with a single matrix whose columns are the components of each vector in the current basis of the space:

[W] = $\begin{bmatrix} w_{11} & w_{12} & \cdots & w_{1n} \\w_{21} & w_{22} & \cdots & w_{2n} \\ \vdots \\ w_{n1} & w_{n2} & \cdots & w_{nn}\end{bmatrix}$.

With this matrix we can write in a relaxed way:

$[\vec w_1, \ldots, \vec w_n] = [\vec v_1, \ldots, \vec v_n]\begin{bmatrix} w_{11} & w_{12} & \cdots & w_{1n} \\w_{21} & w_{22} & \cdots & w_{2n} \\ \vdots \\ w_{n1} & w_{n2} & \cdots & w_{nn}\end{bmatrix} \qquad (2)$

The new components of the same vector u will be called [u']. Substituting (2) inside its expression we have:

$\vec u = u_1' \vec w_1+ \ldots + u_n' \vec w_n = [\vec w_1, \ldots, \vec w_n]\begin{bmatrix} u_1' \\u_2' \\ \vdots \\ u_n'\end{bmatrix} = [\vec v_1, \ldots, \vec v_n]\begin{bmatrix} w_{11} & w_{12} & \cdots & w_{1n} \\w_{21} & w_{22} & \cdots & w_{2n} \\ \vdots \\ w_{n1} & w_{n2} & \cdots & w_{nn}\end{bmatrix}\begin{bmatrix} u_1' \\u_2' \\ \vdots \\ u_n'\end{bmatrix}$

Therefore comparing the two previous expressions and inverting the matrix:

$\begin{bmatrix} u_1 \\u_2 \\ \vdots \\ u_n\end{bmatrix} = \begin{bmatrix} w_{11} & w_{12} & \cdots & w_{1n} \\w_{21} & w_{22} & \cdots & w_{2n} \\ \vdots \\ w_{n1} & w_{n2} & \cdots & w_{nn}\end{bmatrix}\begin{bmatrix} u_1' \\u_2' \\ \vdots \\ u_n'\end{bmatrix} \Rightarrow \begin{bmatrix} u_1' \\u_2' \\ \vdots \\ u_n'\end{bmatrix} = \begin{bmatrix} w_{11} & w_{12} & \cdots & w_{1n} \\w_{21} & w_{22} & \cdots & w_{2n} \\ \vdots \\ w_{n1} & w_{n2} & \cdots & w_{nn}\end{bmatrix}^{-1}\begin{bmatrix} u_1 \\u_2 \\ \vdots \\ u_n\end{bmatrix}$

This shows that vectors change with the inverse of the basis matrix.

Tensor proof

We will prove that the components of a vector change with the inverse of the basis. Let be the N vectors of the basis W a linear span of those of V. Using Einstein notation we can write:

$\vec w_i = w_i^j \vec v_j \qquad (1)$

where W is the matrix of the basis, with the components of its vectors in columns [W] = $\begin{bmatrix} w_1^1 & w_2^1 & \cdots & w_n^1 \\w_1^2 & w_2^2 & \cdots & w_n^2 \\ \vdots \\ w_1^n & w_2^n & \cdots & w_n^n\end{bmatrix}$.

A vector u will have components [u] and [u'] in both bases. Substituting (1) in the second and comparing both:

$\vec u = u^j \vec v_j$

$\vec u = u'^i \vec w_i = u'^i w_i^j \vec v_j$

it follows that that:

$u^j=u'^i w_i^j$

And representing the elements of the inverse of W with a tilde over them:

$u'^i = u^j \tilde w_j^i$

As we wanted to prove.

Change of basis of a linear map

Given a linear map whose matrix is A, in the basis B of the space in transforms vectors coordinates [u] as [v]=A⋅[u]. As vectors change with the inverse of B, its inverse transformation is [v]=B⋅[v'].

Substituting this in the first expression:

$B \cdot [v'] = A \cdot B \cdot [u'] \rightarrow [v']=B^{-1}AB[u']$

Therefore the matrix in the new basis is

$\hat A = B^{-1} \cdot A \cdot B$

being B the matrix of the given basis.

Therefore linear maps are said to be 1-co 1-contra variants objects.

Using tensor notation, with a hat for the new matrix and tildes for the B-inverted elements, the new matrix of components is:

$\hat A^i_j = \tilde b^i_l b^k_j A^l_k$

Change of basis for a general tensor

Let W be the matrix of a new basis, with the components of its vectors in columns [W] = $\begin{bmatrix} w_1^1 & w_2^1 & \cdots & w_n^1 \\w_1^2 & w_2^2 & \cdots & w_n^2 \\ \vdots \\ w_1^n & w_2^n & \cdots & w_n^n\end{bmatrix}$.

As it was shown before, representing the elements of the inverse of W with a tilde over them, and the new components with a hat:

$\hat u^i = \tilde w_j^i u^j$

and for a linear map:

$\hat A^i_j = \tilde w^i_l w^k_j A^l_k$

The components of a general tensor transform in a similar manner with a transformation matrix for each index. If an index transforms like a vector with the inverse of the basis transformation, it is called contravariant and is traditionally denoted with an upper index, while an index that transforms with the basis transformation itself is called covariant and is denoted with a lower index. The transformation law for a rank m tensor with n contravariant indices and m−n covariant indices is thus given as,

$\hat{T}^{i_1,\ldots,i_n}_{i_{n+1},\ldots,i_m}= (W^{-1})^{i_1}_{j_1}\cdots(W^{-1})^{i_n}_{j_n} W^{j_{n+1}}_{i_{n+1}}\cdots W^{j_{m}}_{i_{m}}T^{j_1,\ldots,j_n}_{j_{n+1},\ldots,j_m}.$

Such a tensor is said to be of order or type (n,m−n).^{[Note 1]}

Contravariance and covariance considerations

We have shown that the elements of a linear space are contravariant objects. They change with the inverse of the basis matrix. Not every object of a space behaves this way. For example, linear functionals are covariant objects. Their components change directly with the basis matrix.

Other objects of the space have different variance. For instance, a linear map is a 1-covariant, 1-contravariant object.

Now, if a vector u is covariant, its components do not change with basis, [u]_w = [u]_v. However, they do when looking from the fixed basis. For instance, as the components switch W → (WP = V), the components in basis W leap [u]_w → P[u]_w:

V([u]_v) = WP([u]_w) = W(P[u]_w) ≠ W[u]_w.

More often, a passive transformation is meant. The vectors in the space are considered contravariant, which are preserved, u = V[u]_v = W[u]_w, by compensating the basis transformation through the opposite transformation. That is, as basis scales up by P in W → (V = WP), the components must be scaled down by the same factor, [u]_w → (P⁻¹[u]_w = [u]_v). The net identity transform, I = PP⁻¹, is applied to the vector:

W[u]_w = W(PP^-1)[u]_w = (WP)(P^-1[u]_w) = V[u]_v.

Applying the basis transformation matrix on the left, W → PW, is not widespread, though it can be reduced to the considered W → WP case by PW = (WW⁻¹)PW = W(W⁻¹PW) = WP'.

The change of basis can be applied not only to the component representations of vectors but, at higher level, to the matrices which are component representations of linear transformations.

Linear transformations

The translation of vector coordinates due to basis change is formally equivalent to active linear transformation; both are represented by the same square matrices.^[1] If a linear transformation, L, takes vector x from one space with bases X1 and X2 to a vector y = Lx in another space with bases Y1 and Y2, it then can be expressed in the matrix forms

[y]_Y1 = [L]_Y1X1 [x]_X1 or

[y]_Y2 = [L]_Y2X2 [x]_X2

Since the bases are related by some coordinate transforms, [y]_Y2 = P_Y2Y1 [y]_Y1 and [x]_X2 = P_X2X1 [x]_X1, the first form can be rewritten

P^-1_Y2Y1 [y]_Y2 = [L]_Y1X1 P^-1_X2X1 [x]_X2.

Multiplying both sides by P_Y2Y1

[y]_Y2 = P_Y2Y1 [L]_Y1X1 P^-1_X2X1 [x]_X2

allows to compare with the second form:

[L]_Y2X2 = P_Y2Y1 [L]_Y1X1 P^-1_X2X1.

This relationship is known as equivalence transformation and matrix [L]_Y2X2 is equivalent to [L]_Y1X1 when it holds. A special case of equivalence, called matrix similarity takes place when dimensions of range and domain match and bases are transformed by the same factor, P_Y2Y1 = P_X2X1 = P, resulting

[L]₂ = P [L]₁ P^-1.

The equivalent matrices, therefore, expose the same linear mapping in different bases by making intermediate transition into the old basis, where matrix is known. In the eigenvector basis, the matrix takes the simplest, diagonal, representation.

See also

• Integral transform, the continuous analogue of change of basis.
• Active and passive transformation

External links

• MIT Linear Algebra Lecture on Change of Bases at MIT OpenCourseWare

References

1. ^ D. Aleksandrov, A.N. Kolmogórov and M. A. Lavrent'ev, Mathematics, its content, methods, and meaning. MIT Press, 1963., vol. 3, p. 79.

Notes

1. ^ There is a plethora of different terminology for this around. The terms order, type, rank, valence, and degree are in use for the same concept. This article uses the term "order" or "total order" for the total dimension of the array (or its generalisation in other definitions) m in the preceding example, and the term type for the pair giving the number contravariant and covariant indices. A tensor of type (n,m−n) will also be referred to as a "(n,m−n)" tensor for short.