7Transformation groups

IA Vectors and Matrices

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.