Basematriser

Basematriser, or basis matrices, are a convenient way to package a basis of a finite-dimensional vector space into a square matrix. If V is an n-dimensional vector space over a field, and B = {b1,...,bn} is a basis, the basis matrix B is the n×n matrix whose columns are the basis vectors expressed in a fixed reference basis (often the standard basis). In R^n this is B = [b1 ... bn], and B is invertible because the vectors form a basis.

The basis matrix serves as a bridge between coordinate representations. A vector x in V can be written in standard coordinates x_std, and in the B-basis as [x]_B, with the relation x_std = B [x]_B. Equivalently, [x]_B = B^{-1} x_std, and x_std = B [x]_B.

Change of basis: Given two bases B and C, the change-of-basis matrix from B to C is P_{C<-B} = C^{-1} B. Then for any vector x, [x]_C = P_{C<-B} [x]_B. These matrices allow rewriting linear transformations in terms of different bases, often simplifying their matrix representations.

Example: Let B = { (1,2), (-1,1) } in R^2, so B = [[1,-1],[2,1]] with det 3. Then B^{-1} = (1/3) [[1,1],[-2,1]]. For a vector x = (3,5), the coordinates in basis B are [x]_B = B^{-1} x = (1/3)[ [1,1],[-2,1] ] [3,5]^T = (8/3, -1/3). Check: B [x]_B = x.

Applications: Basematriser are widely used in linear algebra, computer graphics, and engineering to change representations, diagonalize operators, and simplify computations by choosing convenient bases.