4. Determine whether each set equipped with the given operations is a vector space. For those that are not vector spaces, identify the vector space axioms that fail. (a) The set of all real numbers with the standard operations of addition and multiplication. (b) The set of all pairs of real numbers of the form (0, y) with the standard operations on IR². (c) The set of all pairs of real numbers of the form (x, y) where y ≥ 0, with the standard operations on IR². (d) The set of all n-tuples of real numbers of the form (x, x, ..., x) with the standard operations on IR". (e) The set of all triples of real numbers with the standard vector addition but with scalar multipli- cation defined by k(x, y, z) = (k²x, k²y, k²z). (f) The set of all 2×2 invertible matrices with the standard matrix addition and scalar multiplication.