I am trying to figure out the exact meaning of the concepts of 4-vector and relativistic tensor in the Minkowski spacetime. In my understanding, a tensor is a map that assigns an array of numbers to each basis in such a way that certain transformation rules apply. A vector can be viewed as a special case. Relativists define a 4-vector or a relativistic tensor as an object that transforms correctly under the Lorentz transformations. So far so good. I pick a basis and assign it an array of numbers. I then pick another basis that can be obtained by a Lorentz transform and compute a new array of numbers. However, what happens if I pick a basis that cannot be obtained through a Lorentz transform? Is the array undefined? Or is it arbitrary? One way or other, can the relativistic tensor (or 4-vector) be called a mathematical tensor (or vector), considering it does not transform correctly under GL(n,R), but only under a subgroup?