Now that we have the Gram-Schmidt process as a tool, we can use it to come up with orthonormal bases.
Any vector space 
with finite dimension 
has a finite basis 
. This is exactly what it means for to have dimension . And now we can apply the Gram-Schmidt process to turn this basis into an orthonormal basis 
.
We also know that any linearly independent set can be expanded to a basis. In fact, we can also extend any orthonormal collection of vectors to an orthonormal basis. Indeed, if 
is an orthonormal collection, we can add the vectors 
to fill out a basis. Then when we apply the Gram-Schmidt process to this basis it will start with 
, which is already normalized. It then moves on to 
, which is orthonormal with , and so on. Each of the 
is left unchanged, and the 
are modified to make them orthonormal with the existing collection.
April 30, 2009
Posted by John Armstrong |
Algebra, Linear Algebra |
5 Comments
