changeofbasis

Change of basis refers to the process of expressing a vector's coordinates relative to one basis in terms of coordinates relative to another basis in the same vector space. Let V be an n-dimensional vector space over a field F, and let B = {b1,...,bn} and C = {c1,...,cn} be bases of V. For a vector x in V, [x]_B denotes the coordinate column of x with respect to B, and [x]_C with respect to C.

If we denote by P the change-of-basis matrix from C to B, its columns are the C-coordinates of the B-basis vectors, that is P = [ [b1]_C ... [bn]_C ]. Then [x]_C = P [x]_B for all x in V, and [x]_B = P^{-1} [x]_C.

A convenient special case occurs when B is the standard basis and C is an arbitrary basis {c1,...,cn}. In that case the matrix P_{B<-C} with columns c1,...,cn expressed in standard coordinates maps C-coordinates to standard coordinates: [x]_B = P_{B<-C} [x]_C, and [x]_C = P_{B<-C}^{-1} [x]_B.

Example: in R^2, let B be the standard basis and C = {c1=(1,1), c2=(-1,1)}. Then P = [ [1,-1], [1,1] ]. For x = (2,3) in standard coordinates, [x]_C = P^{-1} [x]_B = [[0.5,0.5],[-0.5,0.5]] [2,3]^T = (2.5, 0.5)^T, so x = 2.5 c1 + 0.5 c2.

Change of basis is fundamental for expressing linear transformations in different bases, studying diagonalization, and translating geometric problems between coordinate systems.

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