# \ #
# \mbox{1) Note that, by the given on} \ \ W, \mbox{we may describe} \ \mbox{the elements of} \ \ W \ \mbox{as those vectors of} \ \ V \ \mbox{where the} \ \mbox{sum of the coordinates is} \ 0. #
# \ #
# \mbox{2) Now recall that:} #
# \mbox{two vectors belong to the same coset of any subspace} #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \iff #
# \qquad \mbox{their difference belongs to the subspace itself}. #
# \ #
# \mbox{3) Thus to determine membership in the same coset of} \ W, \ \mbox{it is necessary and sufficient to determine if the} \ \mbox{difference of those vectors belong to} \ W :#
# \qquad \vec{v_1}, \ \vec{v_2} \ \in \ \mbox{same coset of} \ W \quad \iff \quad \vec{v_1} - \vec{v_2} \ \in \ W. #
# \ #
#\mbox{Hence, by the description of} \ W \ \mbox{in (1) above, we have:} #
# \vec{v_1}, \ \vec{v_2} \ \in \ \mbox{same coset of} \ W \quad \iff \quad \mbox{the sum of the coordinates of} \ \ (\vec{v_1} - \vec{v_2}) = 0. #
# \ #
# \mbox{It is a matter of this simple computation.} #
# \ #
# 4) \ \mbox{Proceeding with the two given pairs of vectors, and} \ \mbox{performing this computation on each pair, we find: #
# \quad \mbox{i)} \ \ (1,3,2) - (2,2,2) = (-1,1,0), \ \mbox{and so} #
# \qquad \qquad \mbox{the sum of the coordinates of} \quad (-1,1,0) = 0. #
# \mbox{Hence:} \qquad \qquad \qquad (1,3,2) \ \mbox{and} \ (2,2,2) #
# \qquad \qquad \qquad \qquad \mbox{belong to the same coset of} \ W . #
# \ #
# \quad \mbox{ii)} \ \ (1,1,1) - (3,3,3) = (2,2,2), \ \mbox{and so} #
# \qquad \qquad \mbox{the sum of the coordinates of} \quad (2,2,2) = 6 \ne 0. #
# \mbox{Hence:} \qquad \qquad \qquad (1,1,1) \ \mbox{and} \ (3,3,3) #
# \qquad \quad \quad \ \ \mbox{do not belong to the same coset of} \ W . #
