1. Recall the definition of a vector space: a set V = {u, v, w, . . .} is called a vector space if it is endowed with two operations: (i) scalar multiplication such that for any o E R and u E V, we have Qu E V; (ii) vector addition such that for any u, v E V, we have u tu e V which satisfy the following properties: (the Greek alphabets refer to real numbers (scalars) and the English alphabets refer to vectors from V) (a) utv= v+ u; ( b) ( utv) + w = u+ (0+ w); (c) a(utv) = aut av; (d) (a + B)u = au + Bu; (e) a(Bu) = (aB)u; (f) there is a 0 such that u + 0 = u for any u E V; (g) for any u E V, there is a -u E V such that u + (-u) = 0; (h) lu = u. Now consider V = R+, the set of positive real numbers. On V we define the following "scalar multiplication" and "vector addition": (i) (scalar multiplication, .) for any o E R and u E V, a . u = uo, i.e. raising u to its usual a power; (note: u" still belongs to V so that the scalar multiplication is a legitimate operation. Hence (i) above is satisfied.) (ii) ( vector addition, @) for any u, v E V, u Ov = uv, i.e. taking the usual multiplication between u and v. (note: uv still belongs to V so that the vector addition is a legitimate operation. Hence (ii) above is satisfied.) Prove that V (endowed with the scalar multiplication and vector addition just defined above) is a vector space by showing that all the above properties (yeah, 8 of them) are satisfied. More explicitly, (a) uOv = vou; (b) (utv) Ow = ue (vow); (c) a . (utv) = a . uba .v; (d) (a + B) . u = a . uOB . u; (e) a(B . u) = (aB) . u; 1 (f) there is a 0 such that u @ 0 = u for any u E V; (g) for any u E V, there is a -u E V such that u @ (-u) = 0; (h) 1 . u = u. (Hint: NOTATION MATTERS.)