7.3. Gram-Schmidt Process: Finding an Orthonormal Basis
An orthonormal basis is a basis that consists of mutually orthogonal vectors of length 1. The standard basis for a space of any dimension is an example of an orthonormal basis. There is a nice algorithm called the Gram-Schmidt process that will always find an orthonormal basis given a non-orthonormal basis. Example: [
,
,
,
] represent a basis and we sould like to find [
,
,
,
] which will represent an orthonormal basis.
The following is the Gram-Schmidt process. The number of steps is equal to the number of vectors in the basis. In this case there are 4 steps:
Step 1:
Step 2: 
Step 3: 
Step 4: 
And the pattern continues for as many vectors you are given.
,,,] represent a basis and we sould like to find [,,,] which will represent an orthonormal basis.The following is the Gram-Schmidt process. The number of steps is equal to the number of vectors in the basis. In this case there are 4 steps:
Step 1: