Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
Last updates: 17 October 2011
The general linear group
Let , let be a commutative ring, and let
be the ring of matrices with entries in .
The general linear group with entries in is the group
with operation given by matrix multiplication.
HW: Note that is the group of units in the ring ,
is the group of units in the ring
HW: Show that .
In this case
by permuting the and the reflecting
and , let
matrix with a 1 in the
entry and all other entries
so that the generate
the group as permutation matrices.
Let be a -basis
of so that
Let be a field. Then the group
is presented by generators
PUT IN THE PICTORIAL VERSION OF THE ELEMENTARY MATRICES
Introduce a pictorial notation
corresponds to the matrix identity
Notes and References
This presentation of the general linear group is the motivation for the definition of
Multiplication by elementary matrices is often called "row reduction".
The first half of Theorem ??? says that
by elementary matrices, a fact which is usually
proved (by row reduction) in a first course
in linear algebra right after the definition of matrix multiplication (see [Ar, ???]). The pictorial
notation for elementary matrices appears in [Th, Sec 4.1.2], where it is very handy for proving
identities in unipotent Hecke algebras.