Assume now that we would like to delete a measurement, and update the least-squares solution accordingly.

Assume now we delete the last measurement, that is, replace (a_m, y_m) by (0, 0). We assume that the matrix obtained after deleting any one of the measurements is still full column rank.

1. Express the solution to the problem after deletion, in terms of the original solution, similar to the formula (6.15). Make sure to explain why any quantities you invert are positive.

2. In the so-called leave-one-out analysis, we would like to efficiently compute all the m solutions corresponding to deleting one of the m measurements. Explain how you would compute those solutions computationally efficiently. Detail the number of operations (flops) needed. You may use the fact that to invert a n x n matrix costs O(n³).

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1 We are given a full column rank matrix A € R",", with rows a i = 1,...,m, and a vector y = Rm, and a solution to the least-squares problem m x* = arg min (ax-yi)² = arg min || Ax − y||2₂. - x x