(Solution) - Repeat Exercise 5 5 13 using the weighted norm v -(2025 Original AI-Free Solution)

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Repeat Exercise 5.5.13 using the weighted norm || v || = v21 + 2v22 + v23 + 3v24.
Exercise 5.5.13
Find the closest point to b = (1, 2, - 1,3)T in the subspace W = span {(1,0,2, l)T, (1, 1,0, l)T, (2, 0, 1, - 1)T} by first constructing an orthogonal basis of W and then applying the orthogonal projection formula (5.64).