This is my question: What is the largest m such that there exist v1, ... ,vm ∈ ℝn such that for all i and j, if 1 ≤ i < j ≤ m, then ≤ vi⋅vj = 0 I found a couple of solutions online. http://mathoverflow.net/questions/31436/largest-number-of-vectors-with-pairwise-negative-dot-product http://math.stanford.edu/~akshay/math113/hw7.pdf (problem 10. But it's basically the same solution as the one in the link above) I kind of understand the contradiction but I don't get why there won't be a contradiction when you take m = n + 1. This is my first course in linear algebra, so far I've learnt about linear independence, subspaces, orthogonality but I'm not very familiar with things like inner product spaces - only dot products. I need someone to explain this in simpler terms.