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4.3 Least Squares Approximations

218 Chapter 4. Least Squares ApproximationsIt often happens thatAxDbhas no solution. The usual reason is:too many matrix has more rows than columns. There are more equations than unknowns(mis greater thann). Thencolumns span a small part ofm-dimensional space. Unless allmeasurements are perfect,bis outside that column space. Elimination reaches animpossible equation and stops. But we can t stop just because measurements include repeat: We cannot always get the erroreDb Axdown to zero. Wheneis zero,xis an exact solution the length ofeis as small as possible,bxis aleast Squares goal in this section is to computebxand use it.

Squared length for any x kAx bk2 DkAx pk2 Ckek2: (2) This isthe lawc 2Da Cb2 for a righttriangle. The vectorAx p inthecolumnspaceis ... vector x on the left side of Figure 4.3 went to b DAx on the right side. In that figure x was split into …

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