Problem (2) Read p. 22 where the sum W1 + W2 of two subspaces W1 and W2 of a vector space V is defined and where the direct sum W1 OW2 of two subspaces Wi and W2 of a vector space V is defined. (a Show that W1 + W2 is a subspace of V. (b) In other texts, we see the direct sum of W1 and W2 defined differently. That is, W1 + W2 is a direct sum, and we write W1 OW2, if every element x in W1 + W2 can be written uniquely as a sum x = u1 + u2 where u1 E W1 and u2 E W2. Show that the two definitions are equivalent. (That is, show that if Win W2 = {0}, then every x E W1 + W2 can be written uniquely as a sum x = u1 + u2 where u E Wi and u2 E W2. Then show that, if every x E W1 + W2 can be written uniquely, then WinW2 = {}.) (c) We can defined the sum and direct sum of more than two subspaces. In the case of the direct sum, W1 OW2 0 . . . OWn, it is not sufficient to require that Win Wj = {0} for each 1