Symmetries and Linear Transformations

Abstract

This chapter provides concrete tools for studying and using symmetries in physics. We start by reviewing vectors and then go on to discuss the linear transformations that correspond to spatial translations, rotations and reflections. Next we describe the deep connection between symmetry operations and linear transformations. The key point is that such linear transformations can most easily be represented as matrices that act via matrix multiplication on vectors. The rules for the multiplication of two matrices emerges naturally from the general properties of symmetry transformations. As particularly relevant examples, we show explicitly how to represent rotations and reflections in two dimensions using matrices. Rotations and reflections are transformations that leave the lengths of vectors unchanged, where the length of a vector in Cartesian coordinates is defined using the Pythagoras theorem. As a prelude to our study of special relativity, we end the chapter by emphasizing that the Pythagoras theorem only applies to flat spaces, such as a blackboard, or a tabletop. Other interesting geometries, each with their own version of the Pythagoras theorem, are not only mathematically possible but actually realized in Nature.