Gram-Schmidt Orthogonalization
Overview
One can immediately check that this new basis is orthogonal, meaning
Instead of doing the Gram-Schmidt orthogonalization by hand, we can get sage to do it for us:
A useful result is that
Intuitively, this tells us that the more orthogonal a set of basis for a lattice is, the shorter it is as the volume must be constant.
Exercises
2) Verify that the output of sage is indeed correct.
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