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Subject: Re: orthonormalizing l.d. vectors

In article , giffnews333@444gmail.com
> Hi all,
> my skills in geometry and linear algebra are not so good, so I apologize
> if my question sounds stupid or inappropriate.
> I have got N vector of lenght V, they are not linearly independent. I
> need to make these vectors orthonormal.
> I don't really know how to do this, since the vectors are not lin. ind.
> (in that case I would apply Gram-Schmidt).

The problem divides into two:

1) Pick a maximal linearly independent set of vectors T from V
2) Use Gram-Schmidt on T

You know how to do 2), here is 1)

Place your vectors V in a matrix A as rows vectors. Convert the matrix
into the Row-Reduced Echelon Form (RREF) using elementary row
operations. Call this matrix R.

The non-zero row vectors in R form a basis for the subspace spanned by
V. Their count is r = rank(A) = rank(R), since the row-operations do not
change the rank.

If you wish to select r vectors from the original set V, map the non-
zero vectors from R using the inverse of the matrix which transformed A
to the RREF form. Select the non-zero vectors and you are done.

Kalle Rutanen


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