If N(f) = X then its dimension is n. Otherwise, there must be some element which is not in the null space of f, and by taking a scalar multiple of it, we may assume that . If x is any element of X then , and . So every element of X is a sum of something in the 1-dimensional space spanned by and something in N(f). You should be able to deduce from this that dim(N(f)) = n–1.