Editing Change of basis (section)

{{Short description|Coordinate change in linear algebra}}
{{Distinguish|Change of base (disambiguation){{!}}Change of base}}
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   | image1    = 3d basis transformation.svg
   | caption1  = 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.
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   | caption2  = A vector represented by two different bases (purple and red arrows).
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In [[mathematics]], an [[ordered basis]] of a [[vector space]] of finite [[dimension (vector space)|dimension]] {{mvar|n}} allows representing uniquely any element of the vector space by a [[coordinate vector]], which is a [[finite sequence|sequence]] of {{mvar|n}} [[scalar (mathematics)|scalar]]s called [[coordinates]]. If two different bases are considered, the coordinate vector that represents a vector {{mvar|v}} on one basis is, in general, different from the coordinate vector that represents {{mvar|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.<ref>{{harvtxt|Anton|1987|pp=221–237}}</ref><ref>{{harvtxt|Beauregard|Fraleigh|1973|pp=240–243}}</ref><ref>{{harvtxt|Nering|1970|pp=50–52}}</ref>

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 [[matrix (mathematics)|matrices]], this formula can be written
:<math>\mathbf x_\mathrm{old} = A \,\mathbf x_\mathrm{new},</math>
where "old" and "new" refer respectively to the initially defined basis and the other basis, <math>\mathbf x_\mathrm{old}</math> and <math>\mathbf x_\mathrm{new}</math> are the [[column vector]]s of the coordinates of the same vector on the two bases. {{anchor|Matrix}}<math>A</math> is the '''change-of-basis matrix''' (also called '''transition matrix'''), which is the matrix whose columns are the coordinates of the new [[basis vector]]s on the old basis.

A change of basis is sometimes called a ''change of coordinates'', although it excludes many [[coordinate transformation]]s.
For applications in [[physics]] and specially in [[mechanics]], a change of basis often involves the transformation of an [[orthonormal basis]], understood as a [[rotation (mathematics)|rotation]] in [[physical space]], thus excluding [[translation (geometry)|translations]].
This article deals mainly with finite-dimensional vector spaces.  However, many of the principles are also valid for infinite-dimensional vector spaces.