Question: 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 \(x, y, \alpha\), and \(\beta\) for which superposition fails, i.e.,

1. The spread of values of the vector, defined as \(f(x)=\max _{k} x_{k}-\min _{k} x_{k}\).
2. The difference of the last element and the first, \(f(x)=x_{n}-x_{1}\).
3. The median of an \(n\) -vector, where we will assume \(n=2 k+1\) is odd. The median of the vector \(x\) is defined as the \((k+1)\) st largest number among the entries of \(x\). For example, the median of \((-7.1,3.2,-1.5)\) is \(-1.5\).
4. The average of the entries with odd indices, minus the average of the entries with even indices. You can assume that \(n=2 k\) is even.
5. Vector extrapolation, defined as \(x_{n}+\left(x_{n}-x_{n-1}\right)\), for \(n \geq 2 .\) (This is a simple prediction of what \(x_{n+1}\) would be, based on a straight line drawn through \(x_{n}\) and \(\left.x_{n-1} .\right)\)

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