7 Transformation groups

We have previously seen that orthogonal matrices are used to transform between

orthonormal bases. Alternatively, we can see them as transformations of space

itself that preserve distances, which is something we will prove shortly.

Using this as the definition of an orthogonal matrix, we see that our definition

of orthogonal matrices is dependent on our choice of the notion of distance, or

metric. In special relativity, we will need to use a different metric, which will

lead to the Lorentz matrices, the matrices that conserve distances in special

relativity. We will have a brief look at these as well.