The Unapologetic Mathematician

Mathematics for the interested outsider

Orthonormal Bases

Now that we have the Gram-Schmidt process as a tool, we can use it to come up with orthonormal bases.

Any vector space V with finite dimension d has a finite basis \left\{v_i\right\}_{i=1}^d. This is exactly what it means for V to have dimension d. And now we can apply the Gram-Schmidt process to turn this basis into an orthonormal basis \left\{e_i\right\}_{i=1}^d.

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 \left\{e_i\right\}_{i=1}^n is an orthonormal collection, we can add the vectors \left\{v_i\right\}_{i=n+1}^d to fill out a basis. Then when we apply the Gram-Schmidt process to this basis it will start with e_1, which is already normalized. It then moves on to e_2, which is orthonormal with e_1, and so on. Each of the e_i is left unchanged, and the v_i are modified to make them orthonormal with the existing collection.

April 30, 2009 Posted by | Algebra, Linear Algebra | 5 Comments