orth_2009_09_28_01

orth_2009_09_28_01 - 4 - 1 Orthogonality S. Lall, Stanford...

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Unformatted text preview: 4 - 1 Orthogonality S. Lall, Stanford 2009.09.28.01 4. Orthogonality Orthogonal sets of vectors Orthogonal matrices range-nullspace orthogonality 4 - 2 Orthogonality S. Lall, Stanford 2009.09.28.01 Norms and Inner Products The norm measures the length of a vector bardbl x bardbl = radicalBig x 2 1 + x 2 2 + + x 2 n = x T x It satisfies the Cauchy-Schwartz inequality | x T y | bardbl x bardblbardbl y bardbl The angle between two vectors is = ( x, y ) = cos 1 x T y bardbl x bardblbardbl y bardbl In particular, x and y are orthogonal if x T y = 0 . Write this as x y 4 - 3 Orthogonality S. Lall, Stanford 2009.09.28.01 Orthonormal set of vectors Set of vectors { u 1 , . . . , u k } R n is normalized if bardbl u i bardbl = 1 , i = 1 , . . . , k ( u i are called unit vectors or direction vectors ) orthogonal if u i u j for i negationslash = j orthonormal if both slang: we say u 1 , . . . , u k are orthonormal vectors but orthonormality (like independence) is a property of a set of vectors, not vectors individually 4 - 4...
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orth_2009_09_28_01 - 4 - 1 Orthogonality S. Lall, Stanford...

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