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Linear or not? Determine whether each of 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) = a T 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.

2.1 Lmear or not? Determine whether each or the rollowing scalarvalued tunctions or nvectors is linear. If it is a linear function, give its inner product representation, i.e., an nvector a for which f(:c) = aTm for all :6. If it is not linear, give specic :13, 3;, oz, and B for which superposition fails, 216., ax + 53)) 7E af(93) + fty)- (a) The spread of values of the vector, dened as f (:13) = max;a sch mink 321,. (b) The difference of the last element and the rst, f (m) = as\" 3:1