comp.graphics.algorithms

Subject: Re: orthonormalizing l.d. vectors

On Tue, 02 May 2006 10:58:11 +0200, giff
wrote:
>I have got N vector of lenght V, they are not linearly independent. I
>need to make these vectors orthonormal.

Let's review a few facts and options.

FACTS
* In a vector space of dimension three, at most three vectors can be
linearly independent.
* Given any number of vectors, their span is the linear subspace of
their linear combinations.
* A list of vectors is a basis precisely when it is a linearly
independent list that spans the entire space.
* If we have an inner product on the space (usually the dot product),
then we can define orthogonality to mean the product of two vectors is
zero, and unit length to mean the product of a vector with itself is
one.
* A basis is orthonormal when each vector has unit length, and each is
orthogonal to all the others.

OPTIONS
* From any list of vectors, not all zero, we can extract a linearly
independent list, in general not unique.
* From any independent list (and an inner product) we can construct an
orthonormal list.

* Given any basis for a subspace, we can add enough independent
vectors to make a basis for the entire space.

Whether we're restricting or extending, a handy computational tool is
the QR decomposition.