Gram-Schmidt Procedure

Purpose

The Gram-Schmidt procedure allows us to transform a set of linearly independent vectors into a set of orthonormal vectors that span the same space spanned by the original set.

Lemma

Let . For any ,

That is, the spanning set stays the same even when one of the elements is expressed as a linear combination of the others.

Proof

Since for any , it follows from the theorem linear combinations of spanning set elements that

and since

(not sure what was done here)

Theorem

Let be a basis for a subspace of of . Define

Then is an orthogonal basis for and for each ,

(see span)

Example

Let with and be a basis for . Since , is not an orthogonal basis for . Let

(see projection)
Then is an orthogonal basis for since it contains two nonzero nonparallel vectors and . We may then normalize the vectors:

Visualization