1. The problem statement, all variables and given/known data Is a set of n-tuples which must respect the conditions of closure under addition and closure under scalar multiplication a vector space or a vector subspace? That is, in a 3-dimensional space, are planes which pass by the origin considered to be subspaces of the 3-dimensinal space in question? Or are they considered to be vector spaces? The place where I was reading about it said that subspace of R n and Euclidean vector space are the same thing, but I'm not sure whether it's true or not. I probably misunderstood something. 2. Relevant equations - 3. The attempt at a solution Posting here, as well as Googling.