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17-linearfitt - LEAST SQUARES AND DATA Math 21b O Knill...

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LEAST SQUARES AND DATA Math 21b, O. Knill Section 5.4 : 2,10,22,34,40,16*,18*, it is ok to use technology (i.e. Mathematica) for 34 or 40 is ok but write down the matrices and the steps. GOAL. The best possible ”solution” of an inconsistent linear systems Ax = b will be called the least square solution . It is the orthogonal projection of b onto the image im( A ) of A . The theory of the kernel and the image of linear transformations helps to understand this situation and leads to an explicit formula for the least square fit. Why do we care about non-consistent systems? Often we have to solve linear systems of equations with more constraints than variables. An example is when we try to find the best polynomial which passes through a set of points. This problem is called data fitting . If we wanted to accommodate all data, the degree of the polynomial would become too large, the fit would look too wiggly. Taking a smaller degree polynomial will not only be more convenient but also give a better picture. Especially important is regression , the fitting of data with lines. The above pictures show 30 data points which are fitted best with polynomials of degree 1, 6, 11 and 16. The first linear fit maybe tells most about the trend of the data. THE ORTHOGONAL COMPLEMENT OF im( A ). Because a vector is in the kernel of A T if and only if it is orthogonal to the rows of A T and so to the columns of A , the kernel of A T
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