Question

12.3 Least angle property of least squares. Suppose the m × n matrix A has linearly independent columns, and b is an m-vector. Let x ATb denote the least squares approximate solution (a) Show that for any n-vector a, (Ax)Tb - (Aa)"(Aâ), i.e., the inner product of Ax and b is the same as the inner product of Ax and Ai. Hint. Use (Ax)b (ATb) and (ATA)2 = ATb (b) Show that when A and b are both nonzero, we have The left-hand side is the cosine of the angle between A? and b. Hint. Apply part (a) with -. c) Least angle propertų of least squares. The choice x = x minimizes the distance between Ax and b. Show that x - î also minimizes the angle between Ax and b. (You can assume that Ax and b are nonzero.) Remark. For any positive scalar α, r = αχ also minimizes the angle between Ax and b.

Answer 1

letA'b denote the least sp xres approfinate Solution AnPb

2 les Theve bive

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