NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION: "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
2017-09-28 by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University
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Theorem: Given a set of 
linearly
independent vectors 
from 
, we can construct an orthonormalset 
which are linear combinations of the original
set and which span the same space.
Proof: We prove the theorem by constructing the desired
orthonormal set 
sequentially from the original set 
. This procedure is known as
Gram-Schmidt orthogonalization.
- Set
.
- Define
as the 
minus the projection of onto 
:
The vector is orthogonal
to by construction. (We subtracted out the part of
that wasn't orthogonal to .)
- Set
(i.e., normalize the result of the
preceding step).
- Define
as the 
minus the projection of onto and 
:
- Normalize:
.
- Continue this process until
has been defined.
The Gram-Schmidt orthogonalization procedure will construct an
orthonormal basis from any set of linearly
independent vectors. Obviously, by skipping the normalization step,
we could also form simply an orthogonal basis. The key ingredient
of this procedure is that each new orthonormalbasis vector is
obtained by subtracting out the projection of the next linearly
independent vector onto the vectors accepted so far in the set. We
may say that each new linearly independent vector 
is projected onto the subspace spanned by the
vectors 
, and any nonzero projection in that
subspace is subtracted out of to make it orthogonal to the entire subspace. In other
words, we retain only that portion of each new vector which points along a new dimension. The first direction is
arbitrary and is determined by whatever vector we choose first (
here). The next vector is forced to be
orthogonal to the first. The second is forced to be orthogonal to
the first two, and so on.
This chapter can be considered an introduction to some of the
most important concepts from linear
algebra. The student is invited to pursue further reading in
any textbook on linear algebra, such as [3].
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Appendix: Matlab Examples
Signal Reconstruction from Projections
General Conditions
Index
