Linear and not-linear functions. Determine whether each the following scalar valued functions of n-vectors is linear. If it is a linear function, give its inner product representation,i.e., an n-vector a for which f(x) = aT x for all x. If it is not linear, give specific x, y, , and for which superposition fails, i.e.,

f(x + y) 6= f(x) + f(y).

(a) The spread of values of the vector, defined as f(x) = maxk xk mink xk.

(b) The difference of the last element and the first, f(x) = xn x1.

(c) The median of an n-vector, defined as the middle value of the sorted vector, when n is odd, and the average of the two middle values in the sorted vector, when n is even.

(d) The average of the entries with odd indices, minus the average of the entries with even indices.

(e) Vector extrapolation, defined as xn + (xn xn1), for n 2. (This is a simple prediction of what xn+1 would be, based on a straight line drawn through xn andxn1.)