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Math416_GramSchmidt

# Math416_GramSchmidt - = v ⊥ 2 k v ⊥ 2 k k v ⊥ 2 k 2 =...

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Math 416 - Abstract Linear Algebra Fall 2011, section E1 Gram-Schmidt orthogonalization Let us illustrate the fact that the Gram-Schmidt orthogonalization process works in any inner product space, not just R n (or C n ). Example: Consider the real inner product space C [0 , 1] := { f : [0 , 1] R | f is continuous } with its usual inner product ( f,g ) = Z 1 0 f ( t ) g ( t ) dt . Apply Gram-Schmidt to the linearly independent collection { v 1 = 1 ,v 2 = t } . Solution: u 1 = v 1 k v 1 k k v 1 k 2 = ( v 1 ,v 1 ) = (1 , 1) = Z 1 0 (1)(1) dt = 1 u 1 = v 1 1 = v 1 = 1 v 2 = v 2 - Proj u 1 ( v 2 ) = v 2 - ( v 2 ,u 1 ) u 1 ( v 2 ,u 1 ) = ( t, 1) = Z 1 0 ( t )(1) dt = ± t 2 2 ² 1 0 = 1 2 v 2 = v 2 - 1 2 u 1 = t - 1 2 u 2
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Unformatted text preview: = v ⊥ 2 k v ⊥ 2 k k v ⊥ 2 k 2 = ( v ⊥ 2 ,v ⊥ 2 ) = ( t-1 2 ,t-1 2 ) = ( t,t )-( t, 1) + 1 4 (1 , 1) = 1 3-1 2 + 1 4 = 1 12 ⇒ u 2 = v ⊥ 2 1 / √ 12 = √ 12( t-1 2 ) = √ 3(2 t-1) The resulting orthonormal basis of Span { 1 ,t } is { 1 , √ 3(2 t-1) } . Remark: If we only want orthogonal vectors, without caring about their norms, then the algorithm outputs the orthogonal basis { 1 ,t-1 2 } . 1...
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