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                   However, the converse is often false. In fact, the equation MX = V may have
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                   no solutions at all, but still have least squares solutions to M MX = M V .
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                      Observe that since M is an m × n matrix, then M is an n × m matrix.
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                   Then M M is an n × n matrix, and is symmetric, since (M M) = M M.
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                   Then, for any vector X, we can evaluate X M MX to obtain a num-
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                   ber. This is a very nice number, though! It is just the length |MX| =
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                   (MX) (MX) = X M MX.
                                               Reading homework: problem 1
                      Now suppose that ker L = {0}, so that the only solution to MX = 0 is
                   X = 0. (This need not mean that M is invertible because M is an n × m
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                   matrix, so not necessarily square.) However the square matrix M M is
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                   invertible. To see this, suppose there was a vector X such that M MX = 0.
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                   Then it would follow that X M MX = |MX| = 0. In other words the
                   vector MX would have zero length, so could only be the zero vector. But we
                   are assuming that ker L = {0} so MX = 0 implies X = 0. Thus the kernel
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                   of M M is {0} so this matrix is invertible. So, in this case, the least squares
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                   solution (the X that solves M MX = MV ) is unique, and is equal to
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                                                            −1
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                                               X = (M M) M V.
                   In a nutshell, this is the least squares method:
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                      • Compute M M and M V .
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                      • Solve (M M)X = M V by Gaussian elimination.
                   Example 153 Captain Conundrum falls off of the leaning tower of Pisa and makes
                   three (rather shaky) measurements of his velocity at three different times.

                                                    t s  v m/s
                                                     1     11
                                                     2     19
                                                     3     31

                                                1
                      Having taken some calculus , he believes that his data are best approximated by
                   a straight line
                                                     v = at + b.
                     1
                      In fact, he is a Calculus Superhero.

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