zj with the preceding ones and zj uj1 x1 uj2 x2

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Unformatted text preview: mal system {z1 , z2 , . . . , zj } with the preceding ones and zj = uj,1 x1 + uj,2 x2 + · · · + uj,j xj , with appropriate coefficients uj,k , so zj ∈ span{xl : l ≤ j }. As a consequence, zk ∈ span{xl : l ≤ j } for k ≤ j and thus span{zl : l ≤ j } ⊂ span{xl : l ≤ j } but since the dimension on both sides is equal, the two spans must be the same. 1 The inductive choice of zj is as follows: z1 = x1 . ￿x 1 ￿ Given an orthonormal system {z1 , z2 , . . . , zk−1 } with the same span as {x1 , x2 , . . . , xk−1 }, then we let y k = x k − ￿ x k , z k − 1 ￿ z k −1 − ￿ x k , z k − 2 ￿ z k −2 − · · · − ￿ x k , z 1 ￿ z 1 . We have by the assumed orthogonality of {z1 , z2 , . . . , zk−1 } that ￿yk , zj ￿ = ￿xk , zj ￿−￿xk , zj ￿ = 0 for j < k . On the other hand yk ￿= 0 because xk is not in the span of the preceding xj ’s, which is by induction assumption...
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This note was uploaded on 01/16/2014 for the course MATH 6304 taught by Professor Bernhardbodmann during the Fall '12 term at University of Houston.

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