View source for Change of basis

{{Short description|Coordinate change in linear algebra}}
{{Distinguish|Change of base (disambiguation){{!}}Change of base}}
{{more citations needed|date=November 2017}}<!-- need multiple textbook references -->
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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.

==Change of basis formula==
Let <math>B_\mathrm {old}=(v_1, \ldots, v_n)</math> be a basis of a [[finite-dimensional vector space]] {{mvar|V}} over a [[field (mathematics)|field]] {{mvar|F}}.{{efn|Although a basis is generally defined as a set of vectors (for example, as a spanning set that is linearly independent), the [[tuple]] notation is convenient here, since the indexing by the first positive integers makes the basis an [[ordered basis]].}}

For {{math|1=''j'' = 1, ..., ''n''}}, one can define a vector {{math|''w''{{sub|''j''}}}} by its coordinates <math>a_{i,j}</math> over <math>B_\mathrm {old}\colon</math>
:<math>w_j=\sum_{i=1}^n a_{i,j}v_i.</math>

Let
:<math>A=\left(a_{i,j}\right)_{i,j}</math>
be the [[matrix (mathematics)|matrix]] whose {{mvar|j}}th column is formed by the coordinates of {{math|''w''{{sub|''j''}}}}. (Here and in what follows, the index {{mvar|i}} refers always to the rows of {{mvar|A}} and the <math>v_i,</math> while the index {{mvar|j}} refers always to the columns of {{mvar|A}} and the <math>w_j;</math> such a convention is useful for avoiding errors in explicit computations.)

Setting <math>B_\mathrm {new}=(w_1, \ldots, w_n),</math> one has that <math>B_\mathrm {new}</math> is a basis of {{mvar|V}} if and only if the matrix {{mvar|A}} is [[invertible matrix|invertible]], or equivalently if it has a nonzero [[determinant]]. In this case, {{mvar|A}} is said to be the ''change-of-basis matrix'' from the basis <math>B_\mathrm {old}</math> to the basis <math>B_\mathrm {new}.</math>

Given a vector <math>z\in V,</math> let <math>(x_1, \ldots, x_n) </math> be the coordinates of <math>z</math> over <math>B_\mathrm {old},</math> and <math>(y_1, \ldots, y_n) </math> its coordinates over <math>B_\mathrm {new};</math> that is 
:<math>z=\sum_{i=1}^nx_iv_i = \sum_{j=1}^ny_jw_j.</math>
(One could take the same summation index for the two sums, but choosing systematically the indexes {{mvar|i}} for the old basis and {{mvar|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
:<math>x_i = \sum_{j=1}^n a_{i,j}y_j\qquad\text{for } i=1, \ldots, n.</math>

In terms of matrices, the change of basis formula is
:<math>\mathbf x = A\,\mathbf y,</math>
where <math>\mathbf x</math> and <math>\mathbf y</math> are the column vectors of the coordinates of {{mvar|z}} over <math>B_\mathrm {old}</math> and <math>B_\mathrm {new},</math> respectively.

''Proof:'' Using the above definition of the change-of basis matrix, one has
:<math>\begin{align}
z&=\sum_{j=1}^n y_jw_j\\
 &=\sum_{j=1}^n \left(y_j\sum_{i=1}^n a_{i,j}v_i\right)\\
 &=\sum_{i=1}^n \left(\sum_{j=1}^n a_{i,j} y_j \right) v_i.
\end{align}</math>

As <math>z=\textstyle \sum_{i=1}^n x_iv_i,</math> the change-of-basis formula results from the uniqueness of the decomposition of a vector over a basis.

== Example ==

Consider the [[Euclidean vector space]] <math>\mathbb R^2</math> and a basis consisting of the vectors <math>v_1= (1,0)</math> and <math>v_2= (0,1).</math> If one [[rotation (mathematics)|rotates]] them by an angle of {{mvar|t}}, one has a ''new basis'' formed by <math>w_1=(\cos t, \sin t)</math> and <math>w_2=(-\sin t, \cos t).</math>

So, the change-of-basis matrix is
<math>\begin{bmatrix}
\cos t& -\sin t\\
\sin t& \cos t
\end{bmatrix}.</math>

The change-of-basis formula asserts that, if <math>y_1, y_2</math> are the new coordinates of a vector <math>(x_1, x_2),</math> then one has
:<math>\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}
\cos t& -\sin t\\
\sin t& \cos t
\end{bmatrix}\,\begin{bmatrix}y_1\\y_2\end{bmatrix}.</math>

That is, 
:<math>x_1=y_1\cos t - y_2\sin t \qquad\text{and}\qquad x_2=y_1\sin t + y_2\cos t.</math>
This may be verified by writing
:<math>\begin{align}
x_1v_1+x_2v_2 &= (y_1\cos t - y_2\sin t) v_1  + (y_1\sin t + y_2\cos t) v_2\\
    &= y_1 (\cos (t) v_1 + \sin(t)v_2) + y_2 (-\sin(t) v_1 +\cos(t) v_2)\\
    &=y_1w_1+y_2w_2.
\end{align}</math>

==In terms of linear maps==
Normally, a [[matrix (mathematics)|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 [[basis (linear algebra)|bases]] of implied vector spaces, and to the fact that the choice of a basis induces an [[linear isomorphism|isomorphism]] between a vector space and {{math|''F''{{sup|''n''}}}}, where {{mvar|F}} is the field of scalars. When only one basis is considered for each vector space, it is worth to leave this isomorphism implicit, and to work [[up to]] an isomorphism. As several bases of the same vector space are considered here, a more accurate wording is required.

Let {{mvar|F}} be a [[field (mathematics)|field]], the set <math>F^n</math> of the [[tuple|{{mvar|n}}-tuples]] is a {{mvar|F}}-vector space whose addition and scalar multiplication are defined component-wise. Its [[standard basis]] is the basis that has as its {{mvar|i}}th element the tuple with all components equal to {{math|0}} except the {{mvar|i}}th that is {{math|1}}.

A basis <math>B=(v_1, \ldots, v_n)</math> of a {{mvar|F}}-vector space {{mvar|V}} defines a [[linear isomorphism]] <math>\phi\colon F^n\to V</math> by 
:<math>\phi(x_1,\ldots,x_n)=\sum_{i=1}^n x_i v_i.</math>
Conversely, such a linear isomorphism defines a basis, which is the image by <math>\phi</math> of the standard basis of <math>F^n.</math>

Let <math>B_\mathrm {old}=(v_1, \ldots, v_n)</math> be the "old basis" of a change of basis, and <math>\phi_\mathrm {old}</math> the associated isomorphism. Given a change-of basis matrix {{mvar|A}}, one could consider it the matrix of an [[endomorphism]] <math>\psi_A</math> of <math>F^n.</math> Finally, define
:<math>\phi_\mathrm{new}=\phi_\mathrm{old}\circ\psi_A</math> 
(where <math>\circ</math> denotes [[function composition]]), and
:<math>B_\mathrm{new}= \phi_\mathrm{new}(\phi_\mathrm{old}^{-1}(B_\mathrm{old})). </math>
A straightforward verification shows that this definition of <math>B_\mathrm{new}</math> is the same as that of the preceding section.

Now, by composing the equation <math>\phi_\mathrm{new}=\phi_\mathrm{old}\circ\psi_A</math> with <math>\phi_\mathrm{old}^{-1}</math> on the left and <math>\phi_\mathrm{new}^{-1}</math> on the right, one gets 
:<math>\phi_\mathrm{old}^{-1} = \psi_A \circ \phi_\mathrm{new}^{-1}.</math>
It follows that, for <math>v\in V,</math> one has 
:<math>\phi_\mathrm{old}^{-1}(v)= \psi_A(\phi_\mathrm{new}^{-1}(v)),</math>
which is the change-of-basis formula expressed in terms of linear maps instead of coordinates.

==Function defined on a vector space==

A [[function (mathematics)|function]] that has a vector space as its [[domain of a function|domain]] is commonly specified as a [[multivariate function]] whose variables are the coordinates on some basis of the vector on which the function is [[function application|applied]].

When the basis is changed, the [[expression (mathematics)|expression]] of the function is changed. This change can be computed by substituting the "old" coordinates for their expressions in terms of the "new" coordinates. More precisely, if {{math|''f''('''x''')}} is the expression of the function in terms of the old coordinates, and if {{math|'''x''' {{=}} ''A'''''y'''}} is the change-of-base formula, then {{math|''f''(''A'''''y''')}} 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.

As the change-of-basis formula involves only [[linear function]]s, many function properties are kept by a change of basis. This allows defining these properties as properties of functions of a variable vector that are not related to any specific basis. So, a function whose domain is a vector space or a subset of it is 
* a linear function,
* a [[polynomial function]],
* a [[continuous function]],
* a [[differentiable function]],
* a [[smooth function]],
* an [[analytic function]],
if the multivariate function that represents it on some basis—and thus on every basis—has the same property.

This is specially useful in the theory of [[manifold]]s, as this allows extending the concepts of continuous, differentiable, smooth and analytic functions to functions that are defined on a manifold.

== Linear maps ==
Consider a [[linear map]] {{math|''T'': ''W'' → ''V''}} from a [[vector space]] {{mvar|W}} of dimension {{mvar|n}} to a vector space {{mvar|V}} of dimension {{mvar|m}}. It is represented on "old" bases of {{mvar|V}} and {{mvar|W}} by a {{math|''m''×''n''}} matrix {{mvar|M}}. A change of bases is defined by an {{math|''m''×''m''}} change-of-basis matrix {{mvar|P}} for {{mvar|V}}, and an {{math|''n''×''n''}} change-of-basis matrix {{mvar|Q}} for {{mvar|W}}.

On the "new" bases, the matrix of {{mvar|T}} is 
:<math>P^{-1}MQ.</math>
This is a straightforward consequence of the change-of-basis formula.

== Endomorphisms ==
[[Endomorphism]]s are linear maps from a vector space {{mvar|V}} to itself. For a change of basis, the formula of the preceding section applies, with the same change-of-basis matrix on both sides of the formula. That is, if {{mvar|M}} is the [[square matrix]] of an endomorphism of {{mvar|V}} over an "old" basis, and {{mvar|P}} is a change-of-basis matrix, then the matrix of the endomorphism on the "new" basis is
:<math>P^{-1}MP.</math>

As every [[invertible matrix]] can be used as a change-of-basis matrix, this implies that two matrices are [[similar matrices|similar]] if and only if they represent the same endomorphism on two different bases.

== Bilinear forms ==
A ''[[bilinear form]]'' on a vector space ''V'' over a [[field (mathematics)|field]] {{mvar|F}} is a function {{math|''V'' × ''V'' → F}} which is [[linear map|linear]] in both arguments. That is, {{math|''B'' : ''V'' × ''V'' → F}} is bilinear if the maps
<math>v \mapsto B(v, w)</math>
and <math>v \mapsto B(w, v)</math>
are linear for every fixed <math>w\in V.</math>

The matrix {{math|'''B'''}} of a bilinear form {{mvar|B}} on a basis <math>(v_1, \ldots, v_n) </math> (the "old" basis in what follows) is the matrix whose entry of the {{mvar|i}}th row and {{mvar|j}}th column is <math>B(v_i, v_j)</math>. It follows that if {{math|'''v'''}} and {{math|'''w'''}} are the column vectors of the coordinates of two vectors {{mvar|v}} and {{mvar|w}}, one has
:<math>B(v, w)=\mathbf v^{\mathsf T}\mathbf B\mathbf w,</math>
where <math>\mathbf v^{\mathsf T}</math> denotes the [[transpose]] of the matrix {{math|'''v'''}}.

If {{mvar|P}} is a change of basis matrix, then a straightforward computation shows that the matrix of the bilinear form on the new basis is
:<math>P^{\mathsf T}\mathbf B P.</math>

A [[symmetric bilinear form]] is a bilinear form {{mvar|B}} such that <math>B(v,w)=B(w,v)</math> for every {{mvar|v}} and {{mvar|w}} in {{mvar|V}}. It follows that the matrix of {{mvar|B}} on any basis is [[symmetric matrix|symmetric]]. This implies that the property of being a symmetric matrix must be kept by the above change-of-base formula. One can also check this by noting that the transpose of a matrix product is the product of the transposes computed in the reverse order. In particular,
:<math>(P^{\mathsf T}\mathbf B P)^{\mathsf T} = P^{\mathsf T}\mathbf B^{\mathsf T} P,</math>
and the two members of this equation equal <math>P^{\mathsf T} \mathbf B P</math> if the matrix {{math|'''B'''}} is symmetric.

If the [[characteristic (algebra)|characteristic]] of the ground field {{mvar|F}} is not two, then for every symmetric bilinear form there is a basis for which the matrix is [[diagonal matrix|diagonal]]. Moreover, the resulting nonzero entries on the diagonal are defined up to the multiplication by a square. So, if the ground field is the field <math>\mathbb R</math> of the [[real number]]s, these nonzero entries can be chosen to be either {{math|1}} or {{math|–1}}. [[Sylvester's law of inertia]] is a theorem that asserts that the numbers of {{math|1}} and of {{math|–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 [[quadric]]s 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 matrix|orthogonal]] change-of-base matrix, that is, a matrix such that <math>P^{\mathsf T}=P^{-1}.</math> Such matrices have the fundamental property that the change-of-base formula is the same for a symmetric bilinear form and the endomorphism that is represented by the same symmetric matrix. The [[Spectral theorem]] asserts that, given such a symmetric matrix, there is an orthogonal change of basis such that the resulting matrix (of both the bilinear form and the endomorphism) is a diagonal matrix with the [[eigenvalues]] of the initial matrix on the diagonal. It follows that, over the reals, if the matrix of an endomorphism is symmetric, then it is [[diagonalizable matrix|diagonalizable]].

== See also ==
* [[Active and passive transformation]]
* [[Covariance and contravariance of vectors]]
* [[Integral transform]], the continuous analogue of change of basis.
* [[Chirgwin-Coulson weights]]&nbsp;&mdash; application in computational chemistry

==Notes==
{{notelist}}

== References ==
{{Reflist}}

==Bibliography==
* {{ citation | last1 = Anton | first1 = Howard | year = 1987 | isbn = 0-471-84819-0 | title = Elementary Linear Algebra | edition = 5th | publisher = [[John Wiley & Sons|Wiley]] | location = New York }}
* {{citation | last1 = Beauregard | first1 = Raymond A. | last2 = Fraleigh | first2 = John B. | title = A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields | location = Boston | publisher = [[Houghton Mifflin Company]] | year = 1973 | isbn = 0-395-14017-X | url-access = registration | url = https://archive.org/details/firstcourseinlin0000beau }}
* {{ citation | last1 = Nering | first1 = Evar D. | title = Linear Algebra and Matrix Theory | edition = 2nd | location = New York | publisher = [[John Wiley & Sons|Wiley]] | year = 1970 | lccn = 76091646 }}

==External links==
* [http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-31-change-of-basis-image-compression/ MIT Linear Algebra Lecture on Change of Basis], from MIT OpenCourseWare
*[https://www.youtube.com/watch?v=1j5WnqwMdCk Khan Academy Lecture on Change of Basis], from Khan Academy

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[[Category:Linear algebra]]
[[Category:Matrix theory]]