15-gramschmidt

15-gramschmidt - GRAM SCHMIDT AND QR FACTORIZATION Math...

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Unformatted text preview: GRAM SCHMIDT AND QR FACTORIZATION Math 21b, O. Knill HOMEWORK: Section 5.2: 2,14,16,34,40,42* GRAM-SCHMIDT PROCESS. Let ~v 1 , ..., ~v n be a basis in V . Let ~u 1 = ~v 1 and ~w 1 = ~u 1 / || ~u 1 || . The Gram-Schmidt process recursively constructs from the already constructed orthonormal set ~w 1 , ..., ~w i- 1 which spans a linear space V i- 1 the new vector ~u i = ( ~v i- proj V i- 1 ( ~v i )) which is orthogonal to V i- 1 , and then normalizing ~u i to to get ~w i = ~u i / || ~u i || . Each vector ~w i is orthonormal to the linear space V i- 1 . The vectors { ~w 1 , .., ~w n } form an orthonormal basis in V . EXAMPLE. Find an orthonormal basis for ~v 1 = 2 , ~v 2 = 1 3 and ~v 3 = 1 2 5 . SOLUTION. 1. ~w 1 = ~v 1 / || ~v 1 || = 1 . 2. ~u 2 = ( ~v 2- proj V 1 ( ~v 2 )) = ~v 2- ( ~w 1 ~v 2 ) ~w 1 = 3 . ~w 2 = ~u 2 / || ~u 2 || = 1 . 3. ~u 3 = ( ~v 3- proj V 2 ( ~v 3 )) = ~v 3- ( ~w 1 ~v 3 ) ~w 1- ( ~w 2 ~v 3 ) ~w 2 = 5 , ~w 3 = ~u 3 / || ~u 3 || = 1...
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This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.

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