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

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

Inmathematics, anordered basis of avector space of finitedimensionn allows representing uniquely any element of the vector space by acoordinate vector, which is asequence ofnscalars calledcoordinates. If two different bases are considered, the coordinate vector that represents a vectorv on one basis is, in general, different from the coordinate vector that representsv on the other basis. Achange 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 thechange-of-basis formula which expresses the coordinates relative to one basis in terms of coordinates relative to the other basis. Usingmatrices, this formula can be written

${\displaystyle {\mathbf{x}}_{\mathrm{o}\mathrm{l}\mathrm{d}}=A{\mathbf{x}}_{\mathrm{n}\mathrm{e}\mathrm{w}},}$

where "old" and "new" refer respectively to the initially defined basis and the other basis,${\displaystyle {\mathbf{x}}_{\mathrm{o}\mathrm{l}\mathrm{d}}}$ and${\displaystyle {\mathbf{x}}_{\mathrm{n}\mathrm{e}\mathrm{w}}}$ are thecolumn vectors of the coordinates of the same vector on the two bases.${\displaystyle A}$ is thechange-of-basis matrix (also calledtransition matrix), which is the matrix whose columns are the coordinates of the newbasis vectors on the old basis.

A change of basis is sometimes called achange of coordinates, although it excludes manycoordinate transformations.For applications inphysics and specially inmechanics, a change of basis often involves the transformation of anorthonormal basis, understood as arotation inphysical space, thus excludingtranslations.This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.

Let${\displaystyle B_{\mathrm{o}\mathrm{l}\mathrm{d}}=(v_1,\dots ,v_n)}$ be a basis of afinite-dimensional vector spaceV over afieldF.^[a]

Forj = 1, ...,n, one can define a vectorw_j by its coordinates${\displaystyle a_{i,j}}$ over${\displaystyle B_{\mathrm{o}\mathrm{l}\mathrm{d}}:}$

${\displaystyle w_j=\sum _{i=1}^n{a}_{i,j}{v}_i.}$

Let

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

be thematrix whosejth column is formed by the coordinates ofw_j. (Here and in what follows, the indexi refers always to the rows ofA and the${\displaystyle v_i,}$ while the indexj refers always to the columns ofA and the${\displaystyle w_j;}$ such a convention is useful for avoiding errors in explicit computations.)

Setting${\displaystyle B_{\mathrm{n}\mathrm{e}\mathrm{w}}=(w_1,\dots ,w_n),}$ one has that${\displaystyle B_{\mathrm{n}\mathrm{e}\mathrm{w}}}$ is a basis ofV if and only if the matrixA isinvertible, or equivalently if it has a nonzerodeterminant. In this case,A is said to be thechange-of-basis matrix from the basis${\displaystyle B_{\mathrm{o}\mathrm{l}\mathrm{d}}}$ to the basis${\displaystyle B_{\mathrm{n}\mathrm{e}\mathrm{w}}.}$

Given a vector${\displaystyle z\in V,}$ let${\displaystyle (x_1,\dots ,x_n)}$ be the coordinates of${\displaystyle z}$ over${\displaystyle B_{\mathrm{o}\mathrm{l}\mathrm{d}},}$ and${\displaystyle (y_1,\dots ,y_n)}$ its coordinates over${\displaystyle B_{\mathrm{n}\mathrm{e}\mathrm{w}};}$ 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 indexesi for the old basis andj for the new one makes clearer the formulas that follows, and helps avoiding errors in proofs and explicit computations.)

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

${\displaystyle x_i=\sum _{j=1}^n{a}_{i,j}{y}_j\text{for }i=1,\dots ,n.}$

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 ofz over${\displaystyle B_{\mathrm{o}\mathrm{l}\mathrm{d}}}$ and${\displaystyle B_{\mathrm{n}\mathrm{e}\mathrm{w}},}$ respectively.

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

As${\displaystyle z=\sum _{i=1}^n{x}_i{v}_i,}$ the change-of-basis formula results from the uniqueness of the decomposition of a vector over a basis.

Consider theEuclidean vector space${\displaystyle {\mathbb{R}}^2}$ and a basis consisting of the vectors${\displaystyle v_1=(1,0)}$ and${\displaystyle v_2=(0,1).}$ If onerotates them by an angle oft, one has anew basis formed by${\displaystyle w_1=(\cos t,\sin t)}$ and${\displaystyle w_2=(-\sin t,\cos t).}$

So, the change-of-basis matrix is

The change-of-basis formula asserts that, if${\displaystyle y_1,y_2}$ are the new coordinates of a vector${\displaystyle (x_1,x_2),}$ then one has

That is,

${\displaystyle x_1=y_1\cos t-y_2\sin t\text{and}{x}_2=y_1\sin t+y_2\cos t.}$

This may be verified by writing

Normally, amatrix represents alinear map, and the product of a matrix and a column vector represents thefunction 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 matrixrepresents a linear map, one refers implicitly tobases of implied vector spaces, and to the fact that the choice of a basis induces anisomorphism between a vector space andFⁿ, whereF 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 workup to an isomorphism. As several bases of the same vector space are considered here, a more accurate wording is required.

LetF be afield, the set${\displaystyle F^n}$ of then-tuples is aF-vector space whose addition and scalar multiplication are defined component-wise. Itsstandard basis is the basis that has as itsith element the tuple with all components equal to0 except theith that is1.

A basis${\displaystyle B=(v_1,\dots ,v_n)}$ of aF-vector spaceV defines alinear isomorphism${\displaystyle \varphi :F^n\to V}$ by

${\displaystyle \varphi (x_1,\dots ,x_n)=\sum _{i=1}^n{x}_i{v}_i.}$

Conversely, such a linear isomorphism defines a basis, which is the image by${\displaystyle \varphi }$ of the standard basis of${\displaystyle F^n.}$

Let${\displaystyle B_{\mathrm{o}\mathrm{l}\mathrm{d}}=(v_1,\dots ,v_n)}$ be the "old basis" of a change of basis, and${\displaystyle {\varphi }_{\mathrm{o}\mathrm{l}\mathrm{d}}}$ the associated isomorphism. Given a change-of basis matrixA, one could consider it the matrix of anendomorphism${\displaystyle {\psi }_A}$ of${\displaystyle F^n.}$ Finally, define

${\displaystyle {\varphi }_{\mathrm{n}\mathrm{e}\mathrm{w}}={\varphi }_{\mathrm{o}\mathrm{l}\mathrm{d}}\circ {\psi }_A}$

(where${\displaystyle \circ }$ denotesfunction composition), and

${\displaystyle B_{\mathrm{n}\mathrm{e}\mathrm{w}}={\varphi }_{\mathrm{n}\mathrm{e}\mathrm{w}}({\varphi }_{\mathrm{o}\mathrm{l}\mathrm{d}}^{-1}(B_{\mathrm{o}\mathrm{l}\mathrm{d}})).}$

A straightforward verification shows that this definition of${\displaystyle B_{\mathrm{n}\mathrm{e}\mathrm{w}}}$ is the same as that of the preceding section.

Now, by composing the equation${\displaystyle {\varphi }_{\mathrm{n}\mathrm{e}\mathrm{w}}={\varphi }_{\mathrm{o}\mathrm{l}\mathrm{d}}\circ {\psi }_A}$ with${\displaystyle {\varphi }_{\mathrm{o}\mathrm{l}\mathrm{d}}^{-1}}$ on the left and${\displaystyle {\varphi }_{\mathrm{n}\mathrm{e}\mathrm{w}}^{-1}}$ on the right, one gets

${\displaystyle {\varphi }_{\mathrm{o}\mathrm{l}\mathrm{d}}^{-1}={\psi }_A\circ {\varphi }_{\mathrm{n}\mathrm{e}\mathrm{w}}^{-1}.}$

It follows that, for${\displaystyle v\in V,}$ one has

${\displaystyle {\varphi }_{\mathrm{o}\mathrm{l}\mathrm{d}}^{-1}(v)={\psi }_A({\varphi }_{\mathrm{n}\mathrm{e}\mathrm{w}}^{-1}(v)),}$

which is the change-of-basis formula expressed in terms of linear maps instead of coordinates.

Afunction that has a vector space as itsdomain is commonly specified as amultivariate function whose variables are the coordinates on some basis of the vector on which the function isapplied.

When the basis is changed, theexpression 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, iff(x) is the expression of the function in terms of the old coordinates, and ifx =Ay is the change-of-base formula, thenf(Ay) 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 nomatrix inversion is needed here.

As the change-of-basis formula involves onlylinear functions, 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,
• apolynomial function,
• acontinuous function,
• adifferentiable function,
• asmooth function,
• ananalytic 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 ofmanifolds, as this allows extending the concepts of continuous, differentiable, smooth and analytic functions to functions that are defined on a manifold.

Consider alinear mapT:W →V from avector spaceW of dimensionn to a vector spaceV of dimensionm. It is represented on "old" bases ofV andW by am×n matrixM. A change of bases is defined by anm×m change-of-basis matrixP forV, and ann×n change-of-basis matrixQ forW.

On the "new" bases, the matrix ofT is

${\displaystyle P^{-1}MQ.}$

This is a straightforward consequence of the change-of-basis formula.

Endomorphisms are linear maps from a vector spaceV 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, ifM is thesquare matrix of an endomorphism ofV over an "old" basis, andP is a change-of-basis matrix, then the matrix of the endomorphism on the "new" basis is

${\displaystyle P^{-1}MP.}$

As everyinvertible matrix can be used as a change-of-basis matrix, this implies that two matrices aresimilar if and only if they represent the same endomorphism on two different bases.

Abilinear form on a vector spaceV over afieldF is a functionV ×V → F which islinear in both arguments. That is,B :V ×V → F is bilinear if the maps${\displaystyle v\mapsto B(v,w)}$and${\displaystyle v\mapsto B(w,v)}$are linear for every fixed${\displaystyle w\in V.}$

The matrixB of a bilinear formB on a basis${\displaystyle (v_1,\dots ,v_n)}$ (the "old" basis in what follows) is the matrix whose entry of theith row andjth column is${\displaystyle B(v_i,v_j)}$. It follows that ifv andw are the column vectors of the coordinates of two vectorsv andw, one has

${\displaystyle B(v,w)={\mathbf{v}}^{\mathsf{T}}\mathbf{B}\mathbf{w},}$

where${\displaystyle {\mathbf{v}}^{\mathsf{T}}}$ denotes thetranspose of the matrixv.

IfP is a change of basis matrix, then a straightforward computation shows that the matrix of the bilinear form on the new basis is

${\displaystyle P^{\mathsf{T}}\mathbf{B}P.}$

Asymmetric bilinear form is a bilinear formB such that${\displaystyle B(v,w)=B(w,v)}$ for everyv andw inV. It follows that the matrix ofB on any basis issymmetric. 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,

${\displaystyle (P^{\mathsf{T}}\mathbf{B}P{)}^{\mathsf{T}}=P^{\mathsf{T}}{\mathbf{B}}^{\mathsf{T}}P,}$

and the two members of this equation equal${\displaystyle P^{\mathsf{T}}\mathbf{B}P}$ if the matrixB is symmetric.

If thecharacteristic of the ground fieldF is not two, then for every symmetric bilinear form there is a basis for which the matrix isdiagonal. 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${\displaystyle \mathbb{R}}$ of thereal numbers, these nonzero entries can be chosen to be either1 or–1.Sylvester's law of inertia is a theorem that asserts that the numbers of1 and of–1 depends only on the bilinear form, and not of the change of basis.

Symmetric bilinear forms over the reals are often encountered ingeometry andphysics, typically in the study ofquadrics and of theinertia of arigid body. In these cases,orthonormal bases are specially useful; this means that one generally prefer to restrict changes of basis to those that have anorthogonal change-of-base matrix, that is, a matrix such that${\displaystyle P^{\mathsf{T}}=P^{-1}.}$ 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. TheSpectral 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 theeigenvalues of the initial matrix on the diagonal. It follows that, over the reals, if the matrix of an endomorphism is symmetric, then it isdiagonalizable.

• Active and passive transformation
• Covariance and contravariance of vectors
• Integral transform, the continuous analogue of change of basis.
• Chirgwin-Coulson weights — application in computational chemistry

• Anton, Howard (1987),Elementary Linear Algebra (5th ed.), New York:Wiley,ISBN 0-471-84819-0
• Beauregard, Raymond A.; Fraleigh, John B. (1973),A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston:Houghton Mifflin Company,ISBN 0-395-14017-X
• Nering, Evar D. (1970),Linear Algebra and Matrix Theory (2nd ed.), New York:Wiley,LCCN 76091646

• MIT Linear Algebra Lecture on Change of Basis, from MIT OpenCourseWare
• Khan Academy Lecture on Change of Basis, from Khan Academy

• Linear algebra
• Matrix theory

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