This lecture is part of an online graduate course on Lie groups.
We define the exponential map for matrix groups and describe its basic properties. (We also sketch two ways to define it for general Lie groups.) We give an example to show that it need not be surjective even for connected groups, and an example to show that for infinite dimensional groups there can be problems with its convergence.
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj53RWBkiHKoOsTw-dGHAoJ-h
