Gram-Schmidt
The Gram-Schmidt process is an algorithm in which you input a basis set of vectors and it outputs a basis set that is orthogonal. We can then normalize that set of vectors, and suddenly, we have an orthonormal set of basis vectors! This is very helpful in quantum computing and other areas of applied math, as an orthonormal basis is usually the best basis for computations and representing vectors with coordinates.
Gram-Schmidt Is a Decomposition Tool
While we won't go into it in this book, the Gram-Schmidt process is used in certain decompositions, so it's good to know from that vantage point too.
Let's look at an example before getting into the nitty-gritty of the actual procedure (which can be dry and dull). Let's say I have a basis for ℂ2, such as the following:
These vectors are not orthogonal, since their inner product does not equal 0:
They are also not normalized. Now, I want...
