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Slides_2010_02_08 - Applied linear algebra and numerical...

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Applied linear algebra and numerical analysis Session 15 Prof. Ulrich Hetmaniuk Department of Applied Mathematics February 9, 2010
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Assignment Read Sections 5.3, 5.4, and 5.5 for Friday 02/12.
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Orthogonality v = e 1 + e 2 v = - u 1 + 2 u 2
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Orthogonality v = e 1 + e 2 v = - u 1 + 2 u 2 Fact Computations in orthogonal coordinate systems are preferred.
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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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Orthogonal bases Let V be a real linear space with an inner product < · , · > . Recall that k u k = < u , u > is a norm. Definition A basis u 1 , ··· , u n of V is orthogonal if < u i , u j > = 0 for all i 6 = j . Definition A basis u 1 , ··· , u n of V is orthonormal if it is orthogonal and k u i k = 1 for all i .
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