Hello! Why do we need to impose a change on the basis vector, when going from a reference frame to another. I understand that the components of the vector and the basis change using inverse matrices (the components use a matrix and the vector basis the inverse). But the transformation condition we impose on the components of the 4-vector, already impoose that the vector has the same length in any frame. So why do we need to make sure the vector looks the same in all frames (i.e. why we bother to apply any transformation to the basis, if we are able to keep the length constant anyway)?