Suppose that V = {(x_1, x_2, x_3)} and W = {(x_1, 0, 0)}. Then the quotient space V/W (read as "V mod W") is isomorphic to {(x_2, x_3)} = R^2. In general, when W is a subspace of a vector space V, the quotient space V/W is the set of equivalence classes [v] where v_1 ~v_2 if v_1 - v_2 element W. By "v_1 is equivalent to v_2 modulo W, " it is meant that v_1 = v_2 + w for some w in W, and is another way to say v_1 ~v_2. In particular, the elements of W represent [0]. Sometimes the equivalence classes [v] are written as cosets v + W. The quotient space is an abstract vector space, not necessarily isomorphic to a subspace of V. However, if V has an inner product, then V/W is isomorphic to W^⊥ = {v:〈v, w〉 = 0 for all w element W}.

coset | orthogonal set | quotient space | vector space