Slides_2010_03_01 - Applied linear algebra and numerical...

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Unformatted text preview: Applied linear algebra and numerical analysis Session 23 Prof. Ulrich Hetmaniuk Department of Applied Mathematics March 1, 2010 Review: Counting the solutions Consider a matrix A ∈ R m × n and a linear system Ax = b . • If b / ∈ R ( A ) , then the system has no solution. • If b ∈ R ( A ) , then there is at least one solution. • If N ( A ) = { } , then x = y and there exists only one solution. • If dim N ( A ) > 0, then there exists an infinite number of solutions. Least-Squares Problem Consider a matrix A ∈ R m × n and a vector b / ∈ R ( A ) . Definition The residual vector r ∈ R m is defined by r = b- Ax . • The least-squares problem is looking for a vector x LS such that the 2-norm of the residual is as small as possible x LS = argmin x ∈ R n k b- Ax k 2 • Note that if b ∈ R ( A ) , a solution to Ax = b is also minimizing the residual norm. Least-Squares Problem • Recall k b- Ax k 2 = s m ∑ i = 1 ( b i- ( Ax ) i ) 2 • Minimizing the 2-norm of the residual is equivalent to...
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Slides_2010_03_01 - Applied linear algebra and numerical...

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