### Gram-Schmidt Orthogonalization

Recall from the end of §5.10 above that an*orthonormal*set of vectors is a set of

*unit-length*vectors that are mutually

*orthogonal*. In other words, orthonormal vector set is just an orthogonal vector set in which each vector has been normalized to unit length .

**Theorem:**Given a set of linearly independent vectors from , we can construct an

*orthonormal*set 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*. First, note that for all , since is linearly dependent on every vector. Therefore, .

- Set .
- Define
as minus the projection of
onto
:
- Set
(
*i.e.*, normalize the result of the preceding step). - Define
as minus the projection of
onto
and
:
- Normalize: .
- Continue this process until has been defined.

*subspace*spanned by the vectors , and any nonzero projection in that subspace is subtracted out of to make the new vector 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 thus to the 2D subspace spanned by them), and so on. This chapter can be considered an introduction to some important concepts of

*linear algebra*. The student is invited to pursue further reading in any textbook on linear algebra, such as [47].

^{5.13}Matlab/Octave examples related to this chapter appear in Appendix I.

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Nth Roots of Unity

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Signal/Vector Reconstruction from Projections