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 - bpe =+ GJ GG 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 - ± TT AAx Ab =
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Let's try it - Example: Find the best solution to the system Ax = b when 11 01 00 A = ⎜⎟ and 2 3 2 b =
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Another example - find the best solution to Ax = b when 40 02 11 A = ⎜⎟ and 2 0 11 b =
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Fitting a Straight Line to s Set of Data Points
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This note was uploaded on 08/10/2011 for the course MATH 2331 taught by Professor Staff during the Spring '08 term at University of Houston.

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

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