Change of basis

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A linear combination of one basis of vectors (purple) obtains new vectors (red). If they are linearly independent, these form a new basis. The linear combinations relating the first basis to the other extend to a linear transformation, called the change of basis.

A vector represented by two different bases (purple and red arrows).

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.^[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. Using matrices, this formula can be written

${\displaystyle \mathbf {x} _{\mathrm {old} }=A\,\mathbf {x} _{\mathrm {new} },}$

where "old" and "new" refer respectively to the initially defined basis and the other basis, ${\displaystyle \mathbf {x} _{\mathrm {old} }}$ and ${\displaystyle \mathbf {x} _{\mathrm {new} }}$ are the column vectors of the coordinates of the same vector on the two bases. ${\displaystyle A}$ is the change-of-basis matrix (also called transition matrix), which is the matrix whose columns are the coordinates of the new basis vectors on the old basis.

A change of basis is sometimes called a change of coordinates, although it excludes many coordinate transformations. 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. However, many of the principles are also valid for infinite-dimensional vector spaces.