I have a question about weights of a basis set with respect to the other basis set of one specific vector space. It seems the weights do not covert linearly when basis sets convert linearly. I've got this question from the video on youtube "linear transformation" Let's consider a vector space V spanned upon field K. Then its element x∈V can be expressed in the form of a linear combination with a given basis set and its corresponding weights. x=Σaixi Now if we were to express a vector x with respect to the new basis x_hati, we would express the same vector as x=Σbix_hati We can derive the relation Λ(xi)=x_hati bi=Λ-1ai I did this to cylindrical coordinate, and because https://wikimedia.org/api/rest_v1/media/math/render/svg/cf553bbb290f2b6ad76c9cce12f8807d43ab09ee and according to the equation from video bi=Λ-1ai inverted matrix (transpose of matrix above applied to <x,y,z> should give converted weight bi but this isn't the case. rho=sqrt(x^2+y^2) and phi=tan^-1(y/x) and they are not linear functions. could anyone tell me why is it so?