3 Consider the following possible alternatives to the vector norms we encountered in the notes. For each, state whether the proposed definition satisfies the conditions of a norm or not. If it does, prove it, if it does not, then provide a counter example. (i) The 'median norm': (ii) The 'min-max' norm: (iii) The 'zero' norm: ||||med median{v1, v2],... Un]} ||||mm min |vj|+ max |vk| ||v||2 = 0 k (iv) [Optional] The 'A-weighted' 2-norm, where A is symmetric positive definite: ||||A = v Av 1 The linear system below arises in the determination the location of a GPS using satellite signals. Both the matrix and the right hand-side include measurements involving the timing of signals received from each of the satellites. The atomic clocks on these satellites are accurate to around 10-9 seconds, and both the matrix and the right hand-side will therefore typically have relative errors on this order of magnitude. Given these expected perturbations, that the 2-norm condition number of the matrix below is around 1.46 x 103, and that typical values of ||[x, y, z, d]||2 are around 6700km, estimate the relative accuracy of the GPS system. 0.030 0.006 0.028 0.012 0.038 0.003 0.021 0.013 0.034 0.026 0.014 0.014 0.038 0.001 0.024 0.013 X 177.7 132.7 2 88.7 d 151.8