2331_Notes_4o3_fill - Math 2331 Linear Algebra Section 4.3...

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Math 2331 Linear Algebra Section 4.3 Least Squares Approximation When does Ax = b have a solution? What if there is no exact solution to Ax = b, but you need the "closest" approximation to x for an application. What do you do? The projection of b into the column space - b p e = + G JG G where p = And e = The projection p is the "closest" vector to the column space. Thus, e is as small as possible. So, if we find the projection of b into the column space, (i.e. find p) and then solve ± Ax p = , the vector ± x is the best approximate solution to Ax = b. How do we find p? From the last section - ± T T A Ax A b =
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Let's try it - Example: Find the best solution to the system Ax = b when 1 1 0 1 0 0 A = and 2 3 2 b =
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Another example - find the best solution to Ax = b when 4 0 0 2 1 1 A = and 2 0 11 b =
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