[1][2][3] Such a conversion results from the change-of-basis formula which expresses the coordinates relative to one basis in terms of coordinates relative to the other basis.
For applications in physics and specially in mechanics, a change of basis often involves the transformation of an orthonormal basis, understood as a rotation in physical space, thus excluding translations.
This article deals mainly with finite-dimensional vector spaces.
Let be the matrix whose jth column is formed by the coordinates of wj.
is a basis of V if and only if the matrix A is invertible, or equivalently if it has a nonzero determinant.
that is (One could take the same summation index for the two sums, but choosing systematically the indexes i for the old basis and j for the new one makes clearer the formulas that follows, and helps avoiding errors in proofs and explicit computations.)
With above notation, it is In terms of matrices, the change of basis formula is where
the change-of-basis formula results from the uniqueness of the decomposition of a vector over a basis.
then one has That is, This may be verified by writing Normally, a matrix represents a linear map, and the product of a matrix and a column vector represents the function application of the corresponding linear map to the vector whose coordinates form the column vector.
The change-of-basis formula is a specific case of this general principle, although this is not immediately clear from its definition and proof.
When one says that a matrix represents a linear map, one refers implicitly to bases of implied vector spaces, and to the fact that the choice of a basis induces an isomorphism between a vector space and Fn, where F is the field of scalars.
As several bases of the same vector space are considered here, a more accurate wording is required.
of the n-tuples is a F-vector space whose addition and scalar multiplication are defined component-wise.
by Conversely, such a linear isomorphism defines a basis, which is the image by
denotes function composition), and A straightforward verification shows that this definition of
one has which is the change-of-basis formula expressed in terms of linear maps instead of coordinates.
More precisely, if f(x) is the expression of the function in terms of the old coordinates, and if x = Ay is the change-of-base formula, then f(Ay) is the expression of the same function in terms of the new coordinates.
The fact that the change-of-basis formula expresses the old coordinates in terms of the new one may seem unnatural, but appears as useful, as no matrix inversion is needed here.
This is specially useful in the theory of manifolds, as this allows extending the concepts of continuous, differentiable, smooth and analytic functions to functions that are defined on a manifold.
Consider a linear map T: W → V from a vector space W of dimension n to a vector space V of dimension m. It is represented on "old" bases of V and W by a m×n matrix M. A change of bases is defined by an m×m change-of-basis matrix P for V, and an n×n change-of-basis matrix Q for W. On the "new" bases, the matrix of T is This is a straightforward consequence of the change-of-basis formula.
Endomorphisms are linear maps from a vector space V to itself.
A bilinear form on a vector space V over a field F is a function V × V → F which is linear in both arguments.
(the "old" basis in what follows) is the matrix whose entry of the ith row and jth column is
denotes the transpose of the matrix v. If P is a change of basis matrix, then a straightforward computation shows that the matrix of the bilinear form on the new basis is A symmetric bilinear form is a bilinear form B such that
This implies that the property of being a symmetric matrix must be kept by the above change-of-base formula.
If the characteristic of the ground field F is not two, then for every symmetric bilinear form there is a basis for which the matrix is diagonal.
Moreover, the resulting nonzero entries on the diagonal are defined up to the multiplication by a square.
Sylvester's law of inertia is a theorem that asserts that the numbers of 1 and of –1 depends only on the bilinear form, and not of the change of basis.
Symmetric bilinear forms over the reals are often encountered in geometry and physics, typically in the study of quadrics and of the inertia of a rigid body.
In these cases, orthonormal bases are specially useful; this means that one generally prefer to restrict changes of basis to those that have an orthogonal change-of-base matrix, that is, a matrix such that