ChangeofBasisMatrix

A changeofBasisMatrix, in linear algebra, is an invertible matrix used to translate coordinate representations of vectors from one basis to another. Given two ordered bases B = (b1, ..., bn) and C = (c1, ..., cn) for a finite-dimensional vector space V, the change-of-basis matrix from B to C, denoted P_{C<-B}, satisfies [v]_C = P_{C<-B} [v]_B for every vector v in V. Here [v]_B and [v]_C denote the coordinate column vectors of v with respect to bases B and C, respectively.

Construction and interpretation: To form P_{C<-B}, express each vector in B as a linear combination of the vectors in C: b_j = sum_i a_{ij} c_i. The coefficients a_{ij} form the columns of the matrix A = [a_{ij}]. Then P_{C<-B} = A, and its j-th column is the coordinate vector of b_j in the C-basis. This matrix is invertible, and its inverse P_{B<-C} maps C-coordinates back to B-coordinates, i.e., [v]_B = P_{B<-C} [v]_C.

Applications and examples: The change-of-basis matrix is used to convert representations of vectors when switching bases, solving systems with different bases, and in changing coordinates for linear transformations. If C is the standard basis, the matrix P_{C<-B} has as its columns the standard coordinates of b1, ..., bn, so it directly converts B-coordinates to ordinary coordinates. For a vector with B-coordinates [v]_B, multiplying by P_{C<-B} yields its C-coordinates.