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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 firstly 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, and ${\displaystyle A}$ 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.

This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.

Change of basis formula

Let ${\displaystyle B_{\mathrm {old} }=(v_{1},\ldots ,v_{n})}$ be a basis of a finite-dimensional vector space V over a field F.^[a]

For j = 1, ..., n, one can define a vector w_j by its coordinates ${\displaystyle a_{i,j}}$ over ${\displaystyle B_{\mathrm {old} }\colon }$

${\displaystyle w_{j}=\sum _{i=1}^{n}a_{i,j}v_{i}.}$

Let

${\displaystyle A=\left(a_{i,j}\right)_{i,j}}$

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 ${\displaystyle v_{i},}$ while the index j refers always to the columns of A and the ${\displaystyle w_{j};}$ such a convention is useful for avoiding errors in explicit computations.)

Setting ${\displaystyle B_{\mathrm {new} }=(w_{1},\ldots ,w_{n}),}$ one has that ${\displaystyle B_{\mathrm {new} }}$ is a basis of V if and only if the matrix A is invertible, or equivalently if it has a nonzero determinant. In this case, A is said to be the change-of-basis matrix from the basis ${\displaystyle B_{\mathrm {old} }}$ to the basis ${\displaystyle B_{\mathrm {new} }.}$

Given a vector ${\displaystyle z\in V,}$ let ${\displaystyle (x_{1},\ldots ,x_{n})}$ be the coordinates of ${\displaystyle z}$ over ${\displaystyle B_{\mathrm {old} },}$ and ${\displaystyle (y_{1},\ldots ,y_{n})}$ its coordinates over ${\displaystyle B_{\mathrm {new} };}$ that is

${\displaystyle z=\sum _{i=1}^{n}x_{i}v_{i}=\sum _{j=1}^{n}y_{j}w_{j}.}$

(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.)

The change-of-basis formula expresses the coordinates over the old basis in terms of the coordinates over the new basis. With above notation, it is

${\displaystyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j}\qquad {\text{for }}i=1,\ldots ,n.}$

In terms of matrices, the change of basis formula is

${\displaystyle \mathbf {x} =A\,\mathbf {y} ,}$

where ${\displaystyle \mathbf {x} }$ and ${\displaystyle \mathbf {y} }$ are the column vectors of the coordinates of z over ${\displaystyle B_{\mathrm {old} }}$ and ${\displaystyle B_{\mathrm {new} },}$ respectively.

Proof: Using the above definition of the change-of basis matrix, one has

Zdroj:https://en.wikipedia.org?pojem=Change_of_basis
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