in the second case. Thus we must condense them—this is where the equivalence relation comes into play. The trick to building it is to note that given any vector formula_71 in a vector space, it is always possible to represent it as the sum of two other vectors formula_72 and formula_73 not equal to the original. If nothing else, let formula_72 be any vector and then take formula_75—which also shows that if we are given one vector and then a second vector, we can write the first vector in terms of the second together with a suitable third vector (indeed in many ways—just consider scalar multiples of the second vector in the same subtraction.).