# The Unapologetic Mathematician

## 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

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