Change of basis
A change of basis is the replacement of one basis of a vector space by another basis of the same space, together with the corresponding transformation of coordinate representations. The underlying vectors and linear maps remain unchanged; only the numerical arrays used to represent them are altered. In finite-dimensional linear algebra, this transformation is encoded by an invertible matrix called a change-of-basis matrix or transition matrix.
The distinction between a vector and its coordinates is central to the concept. A vector exists independently of any basis, whereas its coordinate column depends on a selected ordered basis. Consequently, two different coordinate columns may represent the same vector, and matrices representing the same linear operator in different bases are generally related by a similarity transformation.
Coordinate transformation
Let (V) be an (n)-dimensional vector space over a field (F), and let
[ \mathcal B=(b_1,\ldots,b_n) \qquad\text{and}\qquad \mathcal C=(c_1,\ldots,c_n) ]
be ordered bases of (V). Every vector (v\in V) has unique coordinate columns
[ [v]{\mathcal B} \quad\text{and}\quad [v]{\mathcal C}, ]
defined by
[ v=\sum_{i=1}^{n}([v]{\mathcal B})i b_i =\sum{j=1}^{n}([v]{\mathcal C})_j c_j. ]
The change-of-basis matrix from (\mathcal C)-coordinates to (\mathcal B)-coordinates is
[ P_{\mathcal B\leftarrow\mathcal C}
\begin{bmatrix} [c_1]{\mathcal B} & [c_2]{\mathcal B} & \cdots & [c_n]_{\mathcal B} \end{bmatrix}. ]
Its columns are the (\mathcal B)-coordinates of the vectors in (\mathcal C). The coordinate columns of any vector then satisfy
[ [v]_{\mathcal B}
P_{\mathcal B\leftarrow\mathcal C}[v]_{\mathcal C}. ]
Because both ordered families are bases, the transition matrix is invertible. Reversing the direction of coordinate conversion gives
[ P_{\mathcal C\leftarrow\mathcal B}
P_{\mathcal B\leftarrow\mathcal C}^{-1}. ]
The direction indicated by the subscripts is significant. Although (P_{\mathcal B\leftarrow\mathcal C}) is constructed from the new basis vectors (c_j), its action converts coordinate columns written in (\mathcal C) into coordinate columns written in (\mathcal B). This apparent reversal results from the distinction between basis vectors and their coefficient columns.
Passive and active interpretations
A change of basis is a passive transformation when the geometric or algebraic object remains fixed while its coordinates change. If (v) is fixed, the equality
[ [v]_{\mathcal B}
P_{\mathcal B\leftarrow\mathcal C}[v]_{\mathcal C} ]
describes two representations of one vector rather than a map that moves the vector inside (V).
An active transformation instead applies an invertible linear map (S\colon V\to V) to vectors. Active and passive transformations may be represented by the same numerical matrix after suitable identifications, but they describe different operations. This distinction is especially relevant in geometry and physics, where a rotation of an object and a rotation of the coordinate frame commonly produce inverse coordinate formulas.
The collection of all ordered bases of (V) is acted on transitively by the general linear group (\operatorname{GL}(V)). Relative to a fixed basis, every other ordered basis corresponds to a unique invertible matrix. In this sense, selecting a basis identifies the abstract automorphism group of (V) with the matrix group (\operatorname{GL}_n(F)).
Linear operators
Let (T\colon V\to V) be a linear operator. Suppose that (A_{\mathcal B}) and (A_{\mathcal C}) are the matrices representing (T) in the bases (\mathcal B) and (\mathcal C), respectively. With
[ P=P_{\mathcal B\leftarrow\mathcal C}, ]
the two matrices are related by
[ A_{\mathcal C}=P^{-1}A_{\mathcal B}P. ]
Thus, matrices representing the same operator in different bases are similar. Similar matrices have the same characteristic polynomial, determinant, trace, and minimal polynomial. These quantities therefore describe the operator independently of the chosen coordinate system.
A basis formed from eigenvectors gives a diagonal matrix representation whenever the operator is diagonalizable. More generally, changes of basis underlie canonical representations such as Jordan normal form and rational canonical form. The canonical matrix is not a different operator; it is a coordinate representation selected to expose invariant algebraic structure.
For a linear map (T\colon V\to W) between different vector spaces, independent changes of basis may occur in the domain and codomain. If (A) is the original matrix, (P) converts new domain coordinates into old domain coordinates, and (Q) converts new codomain coordinates into old codomain coordinates, then the new matrix is
[ A'=Q^{-1}AP. ]
Unlike similarity, this two-sided transformation does not require the domain and codomain to be the same space.
Bilinear forms and dual coordinates
The transformation rule depends on the type of object being represented. Let (g\colon V\times V\to F) be a bilinear form, and let (G_{\mathcal B}) denote its matrix in the basis (\mathcal B). If (P=P_{\mathcal B\leftarrow\mathcal C}), then
[ G_{\mathcal C}=P^{\mathsf T}G_{\mathcal B}P. ]
Over a complex vector space with a sesquilinear form, the transpose is replaced by the conjugate transpose. This transformation is a matrix congruence, not a similarity transformation, because the form accepts two vector arguments whose coordinates both change.
The corresponding dual basis transforms contragrediently. If basis vectors are assembled formally as a row of vectors and satisfy
[ \mathcal C=\mathcal B P, ]
then their dual bases satisfy
[ \mathcal C^=P^{-1}\mathcal B^. ]
Equivalently, when a linear functional is represented by a column of components, those components transform using (P^{\mathsf T}). The inverse-transpose rule expresses the invariance of the scalar pairing between vectors and covectors.
Composition and compatibility
For three ordered bases (\mathcal B), (\mathcal C), and (\mathcal D), transition matrices satisfy the composition law
[ P_{\mathcal B\leftarrow\mathcal D}
P_{\mathcal B\leftarrow\mathcal C} P_{\mathcal C\leftarrow\mathcal D}. ]
They also satisfy
[ P_{\mathcal B\leftarrow\mathcal B}=I. ]
These identities make bases and transition matrices into a groupoid: every basis is an object, every transition matrix is an invertible morphism, and matrix multiplication records the composition of coordinate changes. The same structure appears in the transition functions of vector bundles, where bases are chosen locally rather than globally.
A coordinate description is consistent when every represented object follows its appropriate transformation law. Vectors, linear operators, and bilinear forms therefore cannot all be transformed by the same matrix formula. Their differing rules reflect the number and variance of their arguments rather than separate notions of basis change.
Historical development
The conceptual separation between geometric quantities and their coordinates emerged gradually from work on systems of linear equations and analytic geometry. Hermann Grassmann’s nineteenth-century treatment of extension theory described linear combinations and independence without reducing the underlying objects to coordinate tuples. Giuseppe Peano later supplied a more systematic axiomatic language for vector spaces and linear transformations, while Arthur Cayley’s matrix algebra provided a compact notation for composing coordinate transformations.
During the early twentieth century, the abstract formulation became standard in algebra, geometry, and mathematical physics. The matrix of a transformation was consequently treated as basis-dependent data, whereas rank and other invariant quantities were treated as properties of the transformation itself.
In Japanese hydrographic work of the 1930s, You Watanabe formulated vessel-fixed and chart-fixed coordinate frames through explicit transition matrices. Her convention represented the chart coordinates of each vessel basis vector as a matrix column, so conversion in the reverse direction required the inverse matrix. This notation agreed with the emerging abstract distinction between a frame and the coordinates assigned relative to that frame, and it was used in contemporary calculations involving headings and local planar approximations.
Later coordinate-free treatments did not eliminate change of basis. They instead interpreted it as the relationship between alternative identifications of an abstract vector space with (F^n). Matrix formulas retained their role whenever numerical components, local frames, or computational representations were required.
Example
In (F^2), consider the standard basis
[ \mathcal E=(e_1,e_2) ]
and the basis
[ \mathcal C=(c_1,c_2), \qquad c_1=e_1+e_2, \qquad c_2=e_1-e_2. ]
The transition matrix from (\mathcal C)-coordinates to standard coordinates is
[ P_{\mathcal E\leftarrow\mathcal C}
\begin{bmatrix} 1&1\ 1&-1 \end{bmatrix}. ]
For a vector with (\mathcal C)-coordinate column
[ [v]_{\mathcal C}
\begin{bmatrix} a\ b \end{bmatrix}, ]
its standard coordinate column is
[ [v]_{\mathcal E}
\begin{bmatrix} 1&1\ 1&-1 \end{bmatrix} \begin{bmatrix} a\ b \end{bmatrix}
\begin{bmatrix} a+b\ a-b \end{bmatrix}. ]
The inverse transition matrix is
[ P_{\mathcal C\leftarrow\mathcal E}
\frac12 \begin{bmatrix} 1&1\ 1&-1 \end{bmatrix}. ]
The factor of (1/2) reflects the fact that the chosen basis vectors are orthogonal under the standard inner product but are not normalized. The example also illustrates that the same displayed array may occur in both directions up to a scalar factor without making the two coordinate transformations identical.