Examples of Vector Spaces: Many different types of mathematical objects can be used to form a vector space. For each set described below, try to impose a vector space structure on it by explicitly defining a notion of vector addition and scalar multiplication. Make sure that your definitions are algebraically closed, for example, if you add two elements in the set you must get another element in the set. Whenever a vector space structure is not possible, explain why not.

(a) The set of all n matrices with real components, R n .

(b) The set of all invertible matrices in R nn , GLn(R).

(c) The set of all polynomials in variable x with real coefficients, R[x].

(d) The set of all zero mean random variables having finite variance, V .