Suppose u1, u2 and u3 are a basis for R^3. So there is a unique combination c1.u1 + c2.u2 + c3.u3 = v for every v in R^3. Now suppose I have this bunch of vectors (u1+u2), (u1+u3), (u2+u3) and I would like to know if they also are a basis for R^3. So v = d1.(u1+u2) + d2.(u1+u3) + d3.(u2+u3) and (d1,d2,d3) has to be unique, right? This will come out nicely in this example, I would get 3 different and unique solutions for d1, d2 and d3. Now my question actually is: is it possible (in some other excercise) to have a unique solution for (d1,d2,d3), so that my new bunch of vectors turns out to be a basis, but that d2 has the same value as d3? And what does this actually mean? I've been thinking about this for a while, but can't seem to figure it out.