The Unapologetic Mathematician

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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 »

  1. […] to see that , take an orthonormal basis for . Then we can expand it to an orthonormal basis of . But now I say that is a basis for . Clearly they’re linearly independent, so we just […]

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  3. […] basis of and hit it with to get a bunch of orthonormal vectors in (orthonormal because . Then fill these out to an orthonormal basis of all of . Just set to be the span of all the new basis vectors, which is […]

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  4. […] any form whatsoever. We know that we can do this by picking a basis of and declaring it to be orthonormal. We don’t anything fancy like Gram-Schmidt, which is used to find orthonormal bases for a […]

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  5. […] of class functions also has a nice inner product. Of course, we could just declare the basis to be orthonormal, but that’s not quite what we’re going to do. Instead, we’ll […]

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