where we have used the associative, commutative, and distributive laws to rearrange the first sum into the second. Continuing this way for scalar multiples and all different-length combinations of vectors allows us to build up a vector addition and scalar multiplication on this set of formal expressions, and we call it the free vector space over formula_31, writing formula_41. Note that the elements of formula_31, considered as length-one formal expressions with coefficient 1 out front, form a Hamel basis for this space.