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# Lecture8 - Lecture 8 8.1 The Gram-Schmidt Process(conti Thm...

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1 Lecture 8 8.1 The Gram-Schmidt Process (conti.) Thm: Orthonormal basis (Gram-Schmidt Process) theorem Let   k a a a ..., , , 2 1 be a basis for a subspace W in R n and let ) ..., , , ( 2 1 j j a a a sp W for j = 1, 2, …, k. then there is an othonormal basis   k q q q ..., , , 2 1 for W such that ) ..., , , ( 2 1 j j q q q sp W for j = 1, 2, …, k. Gram-Schmidt process: (to find an orthonormal basis for a subspace W in R n ) 1) find a basis   k a a a ..., , , 2 1 for W; 2) Let 1 1 a v . 3) To construct a vector 2 v that is orthogonal to 1 v by computing the component of 2 a that is orthogonal to the space W 1 spanned by 1 v . 4) To construct a vector 3 v that is orthogonal to 1 v and 2 v by computing the component of 3 a that is orthogonal to the space W 2 spanned by 1 v and 2 v . 5) To construct a vector j v that is orthogonal to   1 2 1 ..., , , j v v v , j v by computing the component of j a that is orthogonal to the space W j - 1 spanned by   1 2 1 ..., , , j v v v . 6)

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Lecture8 - Lecture 8 8.1 The Gram-Schmidt Process(conti Thm...

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