**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.

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