Slides_2010_02_10 - Applied linear algebra and numerical...

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Applied linear algebra and numerical analysis Session 16 Prof. Ulrich Hetmaniuk Department of Applied Mathematics February 9, 2010
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Orthogonality Orthogonality is the formalization of the geometrical property of perpendicularity . The mathematical formalization is possible because of inner products . Bases with orthogonal elements play an essential role in linear algebra. Computations are simpler and less prone to numerical errors when done in an orthogonal coordinate system.
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Gram-Schmidt process In a real linear space equipped with an inner product, the Gram-Schmidt process will convert any arbitrary basis into an orthogonal basis.
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Gram-Schmidt process Consider the vectors a 1 = ± 3 1 ² and a 2 = ± 2 2 ² .
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Consider the vectors a 1 = 1 1 - 1 a 2 = 1 0 2 a 3 = 2 - 2 3 Compute q 1 such that a 1 = α q 1 k q 1 k 2 = 1. Compute q 2 such that a 2 = β q 1 + γ q 2 q T 1 q 2 = 0 and k q 2 k 2 = 1. Compute
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This note was uploaded on 03/31/2010 for the course AMATH 352 taught by Professor Leveque during the Winter '07 term at University of Washington.

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Slides_2010_02_10 - Applied linear algebra and numerical...

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