What is the nature of reflections in linear algebra?

Reflections in linear algebra are linear transformations that reflect points across a given line or plane. The defining feature of reflections is that they can be represented as orthogonal transformations, which means they preserve the length of vectors and the angle between them.

The important aspect of reflections is that they are invertible mappings. To see why reflections are always invertible, consider that applying a reflection twice will return the original point to its initial position. For instance, if a point is reflected across a line, reflecting the result back across the same line will yield the original point. This property implies that there is a unique reverse operation for every reflection, making these transformations bijections—meaning that they have a one-to-one correspondence between the input and output without any loss of information or dimensionality.

Therefore, reflections are classified as always being invertible since they can be reversed by performing the transformation again. This characteristic distinguishes them from other types of linear transformations that may not have inverses, such as projections or certain non-linear transformations.