6.2 (2.02) Let u, v, w be three different vectors in a vector space V. Consider the three spans S = span({u- v}), S = span({u, v, w}) and S3 = span({u+ v,v + w}). (a) Show that S1 S. (b) Show that S3 CS2. (c) For V = R, given an example of u, v, w for which S = S3. (d) For V = R, given an example of u, v, w for which all of S, S2, S3 are different. 6.3 (2.01) Consider the set X of all functions f: R R. (a) If addition on X is defined as (f+g)(x) = f(x) + g(x) and multiplication is defined as (cf)(x) = f(cx), show that X can not be a vector space. (b) If multiplication is instead defined as (cf)(x) = cf(x), and addition is instead defined as (f+g)(x) = f(g(x)) show that X still can not be a vector space. Hint: Show X is not a vector space with examples!