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10. Let V and W be tWo vector spaces over 1R. Recall that the Cartesian product of two sets is dened as VxW={(u,wj leVandwEW}. Dene the set VQW to be the Cart-an product V X W equipped with the following addition and multiplication operations: Addition = (\"11w1l+{\"21w2l=l\"1 + \"2:1914' WE): 501' a\" [1:1, 1013+ {112,102} E V I? W: Multiplication : :1 ~ (9,11!) = {a . Hit] . w}, for all (411,11!) E V a W and a E R. Under these operations, the space V 6 W is called the direct sum of V and W. (a) Showthat VEBWisavector space, and that V${0} and {U}EBIW arelinear subspaces oil/63W. (b) Show that an}.r nite direct sum of 1.nector spaces gives a vector space (Hint: see Appendix 2 of Damiano and Little). (c) Give an example of this construction