"The components of a vector change under a coordinate transformation, but the vector itself does not." ie: V = a*x + b*y = c*x' + d*y' Though the components (and the basis) have changed, V is still = V. Question 1: Is that right? (I'm assuming so, the main Q is below) Tensor rank (according to wolfram) "The total number of contravariant and covariant indices of a tensor." It is commonly said "A vector is a tensor of rank 1" Does this mean (A): T^a, and R_a are tensors of rank one or does it mean (B): V = (T^a)(R_a) is a tensor of rank one? If it is (A), then how can a vector be regarded as a tensor of rank 1, when it is (contravariant components)*(covariant basis) I'm able to do the maths, but the terminology of 'rank' has been bugging me!