Linear operators

Linear operators are linear transformations that map vectors from and to the same vector space. Indeed, reflections, rotations, and projections are all linear operators. In quantum, we put a "hat" or caret on the top of the letter of the linear operator when we want to distinguish it from its representation as a matrix. For instance, all the following linear transformations are linear operators:

Most of the time, it is clear from the context that we are referring to a matrix or a linear operator, so the caret or "hat" is not used.

The following linear transformations are not linear operators:

You probably noticed that all linear operators are represented by square matrices. This leads to all types of special properties that they can have, such as determinants, eigenvalues...