Applying an invertible linear transformation to express vectors, multilinear forms, or linear operators with respect to a different chosen basis, yielding new coordinate representations related by the change-of-basis matrix.

Principle

Coordinates transform contravariantly for vectors and covariantly for dual objects; matrix representations of linear operators transform by similarity (P^{-1}AP) under a basis change P, while intrinsic properties (rank, eigenvalues as a multiset, determinant up to units) remain invariant.

Demonstration

In R^2, switching from the standard basis to a rotated basis by angle θ corresponds to multiplying coordinate column vectors by the rotation matrix R(θ); the matrix of a linear operator in the rotated basis equals R(θ)^{-1} A R(θ).

Misapplication

Treating a change of basis as altering an operator's intrinsic action rather than only its representation; failing to transform covectors, forms or coordinate-dependent objects appropriately (e.g., leaving the dual basis unchanged).

Consequence

Choosing a convenient basis can simplify computations (sparse or canonical forms), reveal invariant subspaces, and permit canonical normal forms (Jordan, rational, orthogonal diagonalization) that clarify structure and facilitate operations like exponentiation of matrices.

Reversal

Applying the inverse change-of-basis matrix restores the original coordinate representation; composition of basis changes corresponds to multiplication of the corresponding invertible matrices.

Boundary

Requires an invertible linear transformation (change-of-basis matrix must be nonsingular). Does not apply to nonlinear coordinate changes or to objects lacking linear structure; care is required when working over rings that are not fields because invertibility may fail.

Semantic Tension

Tension between passive (coordinate change) and active (moving vectors) viewpoints: the same formulas represent either a relabeling of coordinates or an actual transformation of vectors. Also tension between change of basis and similarity classification: similarity classes classify matrices independent of basis, while basis changes give concrete representatives.

Synthesis

Change of basis is the algebraic operation of re-expressing linear data under a different coordinate frame via an invertible matrix; it preserves intrinsic linear structure while altering representations to reveal computational or structural advantages.