The counterpart to subspaces are quotient vector spaces. Given any subspace W ⊆ ⊆ V , the quotient space V / W (" V modulo W ") is defined as follows: as a set, it consists of v + W = { v + w : w ∈ ∈ W } , where v is an arbitrary vector in V . The sum of two such elements v 1 + W and v 2 + W is ( v 1 + v 2 ) + W , and scalar multiplication is given by a ⋅ ⋅ ( v + W ) = ( a ⋅ ⋅ v ) + W . The key point in this definition is that v 1 + W = v 2 + W if and only if the difference of v 1 and v 2 lies in W . This way, the quotient space "forgets" information that is contained in the subspace W .