Slides_2010_02_12 - Applied linear algebra and numerical...

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Applied linear algebra and numerical analysis Session 17 Prof. Ulrich Hetmaniuk Department of Applied Mathematics February 12, 2010
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Who are they?
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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 = 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 q 3 such that a 3 = δ q 1 + ε q 2 + ζ q 3 q T 1 q 3 = 0, q T 2 q 3 = 0, and k q 3 k 2 = 1. q 1 = 1 3 1 1 - 1 q 2 = 1 42 4 1 5 q 3 = 1 14 2 - 3 - 1
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Gram-Schmidt process Consider the basis vectors a 1 , a 2 , a 3 . We want to build the sequence of orthonormal vectors q 1 , q 2 , q 3 such that span ( a 1 ,..., a j ) = span ( q 1 ,..., q j ) j = 1 ,..., 3 . (1) a 1 = r 11 q 1 a 2 = r 12 q 1 + r 22 q 2 q T 1 a 2 = r 12 q T 1
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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_12 - Applied linear algebra and numerical...

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