LEAST SQUARES AND DATAMath 21b, O. KnillSection 5.4 : 2,10,22,34,40,16*,18*, it is ok to use technology (i.e. Mathematica) for 34 or 40 isok but write down the matrices and the steps.GOAL. The best possible ”solution” of an inconsistent linear systemsAx=bwill be called theleast squaresolution. It is the orthogonal projection ofbonto the image im(A) ofA. The theory of the kernel and the imageof linear transformations helps to understand this situation and leads to an explicit formula for the least squarefit. Why do we care about non-consistent systems? Often we have to solve linear systems of equations with moreconstraints than variables. An example is when we try to find the best polynomial which passes through a set ofpoints. This problem is calleddata fitting. If we wanted to accommodate all data, the degree of the polynomialwould become too large, the fit would look too wiggly. Taking a smaller degree polynomial will not only bemore convenient but also give a better picture. Especially important isregression, 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. Thefirst linear fit maybe tells most about the trend of the data.THE ORTHOGONAL COMPLEMENT OFim(A). Because a vector is in the kernel ofATif and only ifit is orthogonal to the rows ofATand so to the columns ofA, the kernel ofAT
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