The purpose of these notes is to give a concrete introduction to Lie groups and Lie algebras.
Our ulterior motive is to present some beautiful mathematical concepts that can also be
used as tools for solving practical problems arising in computer science, more specifically in
robotics, motion planning, computer vision, computer graphics. Most texts on Lie groups
and Lie algebras begin with prerequesites in diflferential geometry that are often formidable
to average computer scientists (or average scientists, whatever that means!). We have also
banged our head against the wall for a long time, trying to figure out what Lie groups and
Lie algebras are all about, but recently, we realized that there is perhaps a way to break
down the obstacles. We claim that one can sneak into the wonderful world of Lie groups and
Lie algebras by playing with explicit matrix groups such as the group of rotations in R^ (or
R^), and with the exponential map. After actually computing the exponential ^ = e^ of a
2x2 skew symmetric matrix B and observing that it is a rotation matrix, and similarly for
a 3 X 3 skew symmetric matrix S, one begins to suspect that there is something deep going
on. Similarly, after the discovery that every real invertible n x n matrix A can be written
as ^ = RP^ where R is an orthogonal matrix and P is a positive definite symmetric matrix,
and that P can be written as P = e'^ for some symmetric matrix S^ one begins to appreciate
the exponential map.