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Concept
Change of basis
Summary
In mathematics, an ordered basis of a vector space of finite dimension n allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of n scalars called coordinates. If two different bases are considered, the coordinate vector that represents a vector v on one basis is, in general, different from the coordinate vector that represents v on the other basis. A change of basis consists of converting every assertion expressed in terms of coordinates relative to one basis into an assertion expressed in terms of coordinates relative to the other basis.
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. Using matrices, this formula can be written
where "old" and "new" refer respectively to the firstly defined basis and the other basis, and are the column vectors of the coordinates of the same vector on the two bases, and is the change-of-basis matrix (also called transition matrix), which is the matrix whose columns are the coordinate vectors of the new basis vectors on the old basis.
This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.
Let be a basis of a finite-dimensional vector space V over a field F.
For j = 1, ..., n, one can define a vector w_j by its coordinates over
Let
be the matrix whose jth column is formed by the coordinates of w_j. (Here and in what follows, the index i refers always to the rows of A and the while the index j refers always to the columns of A and the such a convention is useful for avoiding errors in explicit computations.)
Setting one has that is a basis of V if and only if the matrix A is invertible, or equivalently if it has a nonzero determinant.
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