Change Of BasisEdit
Change Of Basis
Change of basis is the process by which we translate the description of a vector from one coordinate system to another within a fixed vector space. A vector is a geometric object with a location and direction that does not depend on how we label or measure it. Its coordinates, however, do depend on the chosen basis. Different bases give different coordinate descriptions, even though the underlying vector remains the same. This duality—an invariant object described in different frames—drives both theory and practice in mathematics, physics, and applied disciplines.
In practical terms, selecting a basis is choosing a lens through which problems look simplest or most tractable. One basis may reveal a diagonal or nearly diagonal representation of a linear map, making powers and compositions easier to compute. Another basis might align with measured data, physical frames of reference, or hardware constraints. The mechanism that translates coordinates between these lenses is the change-of-basis matrix. This matrix is invertible whenever both bases are valid, so the move from one description to another is always reversible.
From a pragmatic viewpoint, change of basis underpins everything from computer graphics to statistics to control systems. It lets engineers rotate and scale data, align models with observed phenomena, and compare results across different modeling choices. It also sits at the intersection of concrete computations and abstract structure: even as coordinates shift, essential properties—such as whether a set of vectors spans the space, or whether a map is invertible—remain intact.
Core concepts
Vector spaces, bases, and coordinate representations
A vector space is a collection of objects that can be added together and scaled by numbers from a field, usually the real numbers in introductory contexts. A basis is a finite set of vectors that are linearly independent and span the space. Any vector can be written uniquely as a linear combination of basis vectors. The coefficients in this linear combination are the coordinates of the vector relative to that basis. For a basis B = {b1, ..., bn}, a vector x can be expressed as x = a1 b1 + ... + an bn, and the coordinate vector relative to B is [x]_B = (a1, ..., an).
Change of basis matrix
Let B = {b1, ..., bn} and C = {c1, ..., cn} be two bases for the same space. The change-of-basis matrix from C to B, usually denoted P_{B<-C}, is the matrix whose columns are the coordinates of the C basis vectors expressed in the B basis; equivalently, it converts C-coordinates to B-coordinates. Formally, [x]B = P{B<-C} [x]C for any vector x. The inverse matrix P{C<-B} = P_{B<-C}^{-1} converts the other way: [x]C = P{C<-B} [x]_B.
A common special case is when the two bases are related by a linear map; in that setting, change of basis matrices encode how a map looks in different frames. If A is a linear map represented in basis B and A' is A represented in basis C, they are related by A' = P_{B<-C}^{-1} A P_{B<-C}. This is a form of similarity transformation that preserves many intrinsic features of the map, such as eigenvalues.
Examples and computations
A simple, concrete example helps. Take V = R^2 with the standard basis e1 = (1,0), e2 = (0,1). Let C be the basis consisting of u1 = (1,1) and u2 = (-1,2). The matrix P_{B<-C} that converts coordinates from C to the standard basis B has columns [u1]B and [u2]_B, so P{B<-C} = [[1, -1], [1, 2]]. If a vector x has C-coordinates [x]C = (3, -1), its standard coordinates are [x]_B = P{B<-C} [x]C = (4, 5). Conversely, [x]_C = P{B<-C}^{-1} [x]_B, which you can also compute directly from the definitions.
Invariants and coordinate-free viewpoints
Not all properties depend on the chosen basis. Linear independence, span, and dimension are intrinsic to the space, while coordinates describe the same objects in a particular frame. In more advanced contexts, many results are stated in a coordinate-free language to highlight these invariants. Yet for computation, problem-solving, and intuition, switching to a convenient basis and using a change-of-basis matrix remains a central tool.
Computational aspects
In numerical linear algebra, changing basis is more than a formal exercise. The conditioning of a basis, the stability of the matrix inverse, and the sparsity of matrices all influence efficiency and accuracy. Some basis choices lead to well-conditioned representations that make iterative methods converge faster, while others can amplify rounding errors or waste memory. In practice, the right basis can turn a difficult problem into a straightforward calculation, especially in simulations, optimizations, and data analysis.
Pedagogical debates and practical considerations
There is an ongoing discussion about how best to teach change of basis and related ideas. A coordinate-based approach—building intuition through explicit matrices and concrete computations—tends to prepare students for engineering and applied work, where speed and transparency of steps matter. A coordinate-free or abstract approach emphasizes underlying structure, invariants, and generalization, which can deepen understanding but may feel abstract to those focused on concrete outcomes. In fields where engineering success and real-world results are paramount, practitioners often favor methods that make computations explicit and transferable across contexts, even when that means working through change-of-basis matrices and coordinate representations.
Applications and implications
In computer graphics, changing basis corresponds to rotating, scaling, or transforming coordinates of points and vectors to operate in a camera or world frame. See Principal component analysis for a related idea where a data set is rotated to align with principal directions.
In physics, changing basis is routine when moving between reference frames. The mathematics of these transformations underpins classical mechanics and quantum mechanics, where similarity transformations preserve spectra and other invariants.
In statistics and data science, rotating data to a different basis—such as along principal components—recasts the problem in a space where the axes capture the most variance. See Principal component analysis.
In control theory and signal processing, changing basis can simplify system representations, making it easier to design controllers or filters. See State-space representation and Fourier transform for related ideas about representing signals in different bases.