# Change of basis

A change of basis is the conversion of the coordinates of vectors and the matrices of linear maps from one basis of a vector space to another. The vector itself is unchanged; only its description changes. The conversion is carried out with an invertible matrix, the change of basis matrix, whose columns are the new basis vectors written in the old basis.<sup>[1](https://www.imo.universite-paris-saclay.fr/~bruno.duchesne/enseignement/complements_algebre_lineaire/change-of-basis.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | For bases \( \mathcal{B} = (e_i) \) and \( \mathcal{B}' = (e'_j) \) of \( \mathbf{K}^n \), the change of basis matrix \( P \) from \( \mathcal{B} \) to \( \mathcal{B}' \) has entries \( p_{i,j} \) defined by \( e'_j = p_{1,j} e_1 + \dots + p_{n,j} e_n \); its columns are the new basis vectors in the old basis.<sup>[1](https://www.imo.universite-paris-saclay.fr/~bruno.duchesne/enseignement/complements_algebre_lineaire/change-of-basis.html)</sup> |
| Coordinate conversion | If \( X' \) holds the coordinates of a vector \( u \) in \( \mathcal{B}' \), then \( X = PX' \) holds the coordinates of the same \( u \) in \( \mathcal{B} \).<sup>[1](https://www.imo.universite-paris-saclay.fr/~bruno.duchesne/enseignement/complements_algebre_lineaire/change-of-basis.html)</sup> |
| Invertibility | \( P \) is invertible, and the change of basis matrix from \( \mathcal{B}' \) back to \( \mathcal{B} \) is \( P^{-1} \).<sup>[1](https://www.imo.universite-paris-saclay.fr/~bruno.duchesne/enseignement/complements_algebre_lineaire/change-of-basis.html)</sup> |
| Matrix of a linear map | If \( A \) is the matrix of \( f \colon \mathbf{K}^n \to \mathbf{K}^n \) in \( \mathcal{B} \) and \( A' \) its matrix in \( \mathcal{B}' \), then \( A' = P^{-1}AP \) (equivalently \( A = PA'P^{-1} \)).<sup>[1](https://www.imo.universite-paris-saclay.fr/~bruno.duchesne/enseignement/complements_algebre_lineaire/change-of-basis.html)</sup> |
| Interpretation | \( P \) is the matrix of the identity map with respect to the pair of bases \( (\mathcal{B}', \mathcal{B}) \), that is \( P = \mathrm{Mat}_{\mathcal{B}',\mathcal{B}}(\mathrm{id}) \).<sup>[1](https://www.imo.universite-paris-saclay.fr/~bruno.duchesne/enseignement/complements_algebre_lineaire/change-of-basis.html)</sup> |

## The change of basis matrix

Let \( \mathcal{B} = (e_1, \dots, e_n) \) and \( \mathcal{B}' = (e'_1, \dots, e'_n) \) be two bases of \( \mathbf{K}^n \). Each new basis vector is a linear combination of the old ones, \( e'_j = p_{1,j} e_1 + \dots + p_{n,j} e_n \), and the coefficients \( p_{i,j} \) are placed in the columns of a matrix \( P \). This is the change of basis matrix from \( \mathcal{B} \) to \( \mathcal{B}' \), sometimes written \( P_{\mathcal{B}}^{\mathcal{B}'} \).<sup>[1](https://www.imo.universite-paris-saclay.fr/~bruno.duchesne/enseignement/complements_algebre_lineaire/change-of-basis.html)</sup>

**Converting coordinates.** If a vector \( u \) has coordinates \( X' = (X'_1, \dots, X'_n) \) in \( \mathcal{B}' \), so that \( u = \sum_j X'_j e'_j \), then the column vector \( X = PX' \) contains the coordinates of the same \( u \) in \( \mathcal{B} \).<sup>[1](https://www.imo.universite-paris-saclay.fr/~bruno.duchesne/enseignement/complements_algebre_lineaire/change-of-basis.html)</sup> The direction of the multiplication is a common source of confusion: the matrix built from the new basis vectors, expressed in the old basis, converts new coordinates into old coordinates.<sup>[2](https://homepage.physics.uiowa.edu/~pkaaret/2018f_p4905/L19.html)</sup>

Because \( P \) equals the matrix of the identity map \( \mathrm{id} \) with respect to \( \mathcal{B}' \) on the left and \( \mathcal{B} \) on the right, and the identity is a bijection, \( P \) is invertible. The change of basis matrix in the reverse direction is \( P^{-1} \).<sup>[1](https://www.imo.universite-paris-saclay.fr/~bruno.duchesne/enseignement/complements_algebre_lineaire/change-of-basis.html)</sup>

## The change of basis formula

For a linear map \( f \colon \mathbf{K}^n \to \mathbf{K}^n \) with matrix \( A \) in \( \mathcal{B} \) and matrix \( A' \) in \( \mathcal{B}' \), the two matrices are related by

\[ A = P A' P^{-1}, \qquad A' = P^{-1} A P. \]

The formula follows by composing maps: to apply \( f \) to a vector given in \( \mathcal{B}' \) coordinates, first convert to \( \mathcal{B} \) coordinates with \( P \), apply \( A \), and convert back to \( \mathcal{B}' \) coordinates with \( P^{-1} \).<sup>[1](https://www.imo.universite-paris-saclay.fr/~bruno.duchesne/enseignement/complements_algebre_lineaire/change-of-basis.html)</sup> The same relation holds for a map \( T \colon V \to V \) on an abstract vector space, where \( P \) is the matrix of the identity between the two bases.<sup>[3](https://www.lancaster.ac.uk/~prendivs/accessible/math220/MATH220-Complete-Notes-2017.tex/Ch4.S7.html)</sup> For a map \( T \colon V \to W \) between different spaces, with separate changes of basis in \( V \) and \( W \), the identity \( T = \mathrm{id}_W \circ T \circ \mathrm{id}_V \) gives the corresponding four-matrix formula.<sup>[4](https://www.homepages.ucl.ac.uk/~ucahmto/0005_2021/Ch4.S21.html)</sup>

Matrices related by \( A' = P^{-1}AP \) are called similar matrices; they represent the same linear map under different bases and share their eigenvalues, determinant, and rank.

## A worked example

Take \( \mathcal{B} \) as the canonical basis of \( \mathbf{R}^2 \) and \( \mathcal{B}' = (e'_1, e'_2) \) with \( e'_1 = (1,1) \) and \( e'_2 = (1,-1) \). The change of basis matrix and its inverse are

\[ P = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}, \qquad P^{-1} = \begin{bmatrix} \tfrac{1}{2} & \tfrac{1}{2} \\ \tfrac{1}{2} & -\tfrac{1}{2} \end{bmatrix}. \]

A vector with coordinates \( (5,1) \) in the canonical basis has coordinates \( (3,2) \) in \( \mathcal{B}' \), since \( P^{-1}\begin{bmatrix}5\\1\end{bmatrix} = \begin{bmatrix}3\\2\end{bmatrix} \).<sup>[1](https://www.imo.universite-paris-saclay.fr/~bruno.duchesne/enseignement/complements_algebre_lineaire/change-of-basis.html)</sup>

Now let \( f \) have the diagonal matrix \( A' = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} \) in \( \mathcal{B}' \). In the canonical basis its matrix is

\[ A = P A' P^{-1} = \begin{bmatrix} \tfrac{3}{2} & -\tfrac{1}{2} \\ -\tfrac{1}{2} & \tfrac{3}{2} \end{bmatrix}. \]

The matrix is much simpler in \( \mathcal{B}' \) than in the canonical basis, so calculations with \( f \) are advantageously done in \( \mathcal{B}' \).<sup>[1](https://www.imo.universite-paris-saclay.fr/~bruno.duchesne/enseignement/complements_algebre_lineaire/change-of-basis.html)</sup> This is the general principle: a linear transformation that looks complicated in one basis can become easy to understand in the right one, for example a reflection along a line through the origin becomes a simple sign change in a basis aligned with that line.<sup>[5](https://www.homepages.ucl.ac.uk/~ucahmto/0005_2021/MATH0005_lecture_notes.pdf)</sup><sup> • </sup><sup>[6](https://interactivetextbooks.tudelft.nl/linear-algebra/Chapter4/ChangeOfBasis.html)</sup>

## Notation and conventions

Textbooks differ in the direction they assign to the change of basis matrix. The Paris-Saclay notes define \( P \) so that \( X = PX' \) converts new coordinates to old ones; the Delft quantum physics notes define \( P_{B \to C} \) so that its \( j \)-th column holds the coordinates of the \( B \)-basis vector \( \vb_j \) relative to \( C \), giving \( A^{(C)} = P A^{(B)} P^{-1} \); the CUNY notes call the analogous object the transition matrix \( \mathrm{Rep}_B^{B'} \) and write \( [\vec v]_{B'} = \mathrm{Rep}_B^{B'}\,[\vec v]_B \).<sup>[1](https://www.imo.universite-paris-saclay.fr/~bruno.duchesne/enseignement/complements_algebre_lineaire/change-of-basis.html)</sup><sup> • </sup><sup>[7](https://interactivetextbooks.tudelft.nl/mqp-v2/basis-transformations.html)</sup><sup> • </sup><sup>[8](https://pressbooks.cuny.edu/mat5600linearalgebra/chapter/coordinates-and-change-of-basis/)</sup>

## Applications

**Choosing a convenient basis.** The main practical use of a change of basis is to replace a hard matrix by an easy one. Diagonalization, introduced by [Augustin-Louis Cauchy](https://www.edgepedia.org/augustin-louis-cauchy) in 1826 in work on eigenvalues, and the [Jordan normal form](https://www.edgepedia.org/jordan-normal-form) of 1870 are both statements that a suitable basis makes the matrix of a map as simple as the map allows.<sup>[9](https://people.math.harvard.edu/~knill/teaching/math21b2001/history/index.html)</sup>

**Physics.** In tensor language, a change of basis is the prototype of a coordinate transformation. The polarization tensor \( \alpha_{ij} \) of a crystal, with its nine components linking the electric field to the polarization, transforms in a definite way when the coordinate system is changed, while the physical polarization \( \mathbf{P} \) is unchanged; the same transformation rule defines Cartesian tensors generally.<sup>[10](https://www.feynmanlectures.caltech.edu/II_31.html)</sup><sup> • </sup><sup>[11](https://galileoandeinstein.phys.virginia.edu/7010/CM_23_Rigid_Body_Motion.html)</sup>

**Numerical linear algebra.** Similarity transformations are the working tool of eigenvalue software. The LAPACK routine for nonsymmetric eigenproblems optionally applies a balancing transformation, replacing \( A \) by \( D A D^{-1} \) with \( D \) diagonal, to make rows and columns closer in norm and reduce the condition numbers of the computed eigenvalues and eigenvectors.<sup>[12](https://www.netlib.org/lapack/explore-html/d4/d42/zgeevx_8f_source.html)</sup>

**Signal and image compression.** In Gilbert Strang's MIT linear algebra course, change of basis is presented together with image compression: compressing a signal or an image is exactly a change of basis, expressing the data in a basis in which most coefficients are small enough to discard.<sup>[13](https://live.ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/46108efb43dcf33e989eb5e101fcf9b5_MIT18_06S10_L31.pdf)</sup>

## References

1. [Chapter 3: Change of Basis, Linear Algebra Complements, Université Paris-Saclay](https://www.imo.universite-paris-saclay.fr/~bruno.duchesne/enseignement/complements_algebre_lineaire/change-of-basis.html)
2. [Lecture #19, Applied Linear Algebra, University of Iowa](https://homepage.physics.uiowa.edu/~pkaaret/2018f_p4905/L19.html)
3. [4.G Change of basis matrices, MATH220 Linear Transformations, Lancaster University](https://www.lancaster.ac.uk/~prendivs/accessible/math220/MATH220-Complete-Notes-2017.tex/Ch4.S7.html)
4. [4.21 Change of basis, MATH0005 Algebra 1, University College London](https://www.homepages.ucl.ac.uk/~ucahmto/0005_2021/Ch4.S21.html)
5. [MATH0005 lecture notes PDF, University College London](https://www.homepages.ucl.ac.uk/~ucahmto/0005_2021/MATH0005_lecture_notes.pdf)
6. [4.3 Change of basis, Linear Algebra, TU Delft](https://interactivetextbooks.tudelft.nl/linear-algebra/Chapter4/ChangeOfBasis.html)
7. [6. Basis transformations, Mathematics for Quantum Physics, TU Delft](https://interactivetextbooks.tudelft.nl/mqp-v2/basis-transformations.html)
8. [Coordinates and Change of Basis, Linear Algebra: Lecture Notes, CUNY](https://pressbooks.cuny.edu/mat5600linearalgebra/chapter/coordinates-and-change-of-basis/)
9. [About the history of linear algebra, Harvard Math 21b](https://people.math.harvard.edu/~knill/teaching/math21b2001/history/index.html)
10. [The Feynman Lectures on Physics Vol. II Ch. 31: Tensors](https://www.feynmanlectures.caltech.edu/II_31.html)
11. [Rigid Body Motion, University of Virginia](https://galileoandeinstein.phys.virginia.edu/7010/CM_23_Rigid_Body_Motion.html)
12. [LAPACK: SRC/zgeevx.f Source File, Netlib](https://www.netlib.org/lapack/explore-html/d4/d42/zgeevx_8f_source.html)
13. [18.06 Linear Algebra Transcript, Lecture 31, MIT OpenCourseWare](https://live.ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/46108efb43dcf33e989eb5e101fcf9b5_MIT18_06S10_L31.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Linear and multilinear algebra › Vector spaces and linear maps*

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