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Linear and not-linear functions. Determine whether each the following scalar valued functions of n-vectors is linear. If it is a linear function,...
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, α, and β for which superposition fails, i.e.,
f(αx + βy) 6= αf(x) + βf(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 − xn−1), for n ≥ 2. (This is a simple prediction of what xn+1 would be, based on a straight line drawn through xn andxn−1.)
