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15-gramschmidt

# 15-gramschmidt - GRAM SCHMIDT AND QR FACTORIZATION HOMEWORK...

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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 0 0 , ~v 2 = 1 3 0 and ~v 3 = 1 2 5 . SOLUTION. 1. ~w 1 = ~v 1 / || ~v 1 || = 1 0 0 . 2. ~u 2 = ( ~v 2 - proj V 1 ( ~v 2 )) = ~v 2 - ( ~w 1 · ~v 2 ) ~w 1 = 0 3 0 . ~w 2 = ~u 2 / || ~u 2 || = 0 1 0 . 3. ~u 3 = ( ~v 3 - proj V 2 ( ~v 3 )) = ~v 3 - ( ~w 1 · ~v 3 ) ~w 1 - ( ~w 2 · ~v 3 ) ~w 2 = 0 0 5 , ~w 3 = ~u 3 / || ~u 3 || = 0 0 1 . QR FACTORIZATION . The formulas can be written as ~v 1 = || ~v 1 || ~w 1 = r 11 ~w 1 · · · ~v i = ( ~w 1 · ~v i ) ~w 1 + · · · + ( ~w i - 1 · ~v i ) ~w i - 1 + || ~u i || ~w i = r i 1 ~w 1 + · · · + r ii ~w i · · ·
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