Gidon Eshel
491 Hinds
Dept. of the Geophysical
Sciences,
5734 S. Ellis Ave., The Univ. of Chicago,
Chicago, IL
60637
(773) 702-0440, geshel@midway.uchicago.edu
If we have a set of linearly-independent but non-orthogonal vectors , , we
often wish to turn them into an alternative set of vectors, , , that
are mutually orthonormal,
Let's address the particular example of
The first is simple, involving a simple normalization;
The second vector will be the original second vector, , minus
its projection on the first orthonormal basis vector, . That is,
To construct the last basis vector, we need to subtract from its
projections on both and
;
In conclusion, we have transformed our original, non-orthonormal, basis set,