Linear Map - Change of Basis

Change of Basis

Given a linear map whose matrix is A, in the basis B of the space it transforms vectors coordinates as = A. As vectors change with the inverse of B, its inverse transformation is = B.

Substituting this in the first expression

hence

Therefore the matrix in the new basis is A′ = B−1AB, being B the matrix of the given basis.

Therefore linear maps are said to be 1-co 1-contra -variant objects, or type (1, 1) tensors.