A useful form of the quotient rule for three-dimensional vectors is this: Suppose that a and b are known to be three-vectors and suppose that for every orthogonal set of axes there is a 3 x 3 matrix T with the property that b = Ta for every choice of a, then T is a tensor.
(a) Prove this.
(b) State and prove the correspornding rule for four-vectors and four-tensors.