1. The problem statement, all variables and given/known data 1) Is S a subspace of R^n? 1.1) Given n=4 and a vector is in S if it is in the span of e1, e2 or in the span of e3, e4 where e1...e4 is the canonical basis of R^4 1.2) Given n=3 and S is a sphere of radius 1. 2) Let S be a subspace of R^10 with basis v1; v2; v3. Show that the vectors 2v1 + 4v2; v2 + 2v3; 5v3 also form a basis of S. 3. The attempt at a solution 1.1) No, because if the vector only spans e1, e2 or e3, e4, then it would only generate a plane in R^4? 1.2) I was thinking yes because a sphere is a 3-d object, thus would be a subspace of itself in R^3. However, if the radius is one then would that be false because if I were to add the vectors of the sphere, it would be outside the sphere? 2) From what I read, I think I would have to prove that they are linear independent then prove that they can generate R^10 (both of which are already proven if v1,v2,v3 is the basis of R^10) but how would I show it without any actual numbers? Thanks!