Reading in Mathematical Physics by Sadri Hassani. It defines a vector abstractly. I will repeat that definition here rather more informally. There are these things called vectors, a, b, x etc., that have these properties: You can add them a + b = b + a a + (b + c) = (b + a) + c a + 0 = a, 0 is the zero vector a + (- a) = 0 You can multiply them by complex numbers (scalars) like c, d c(d a) = (cd)a 1 a = a Multiplication involving vectors and scalars is distributive c(a + b) = c a+ c b (c + d) a = c a+ d a And that is it. Given that definition, a scalar is a vector, a matrix is a vector, a tensor is a vector. Yes? Mind you, I have also read that scalars and vectors are a kinds of tensors, of rank 0 and 1 respectively. True? Am I confused? Should I be?