Part 7: Linear Groups |
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
A linear representation of O3 |
A linear
group is a group of non-singular linear transformations of a
vector space V. The group of all such transformations is denoted
GL(V).
In this course we are mainly interested in the finite dimensional case,
so let
V a vector space of dimension n over
R. Denote by
Mn the ring of nxn matrices over
R. Then by a
suitable choice of basis, we can assume that
GL(V) = GL(n,R) = {A 1
Mn: A is invertible}
In general a real linear group of degree n is a subgroup of GL(n,R). Consider a cube of side length 2, centered at the origin of R3 with its edges parallel to the coordinate axes. Then each rigid motion of the cube is a rotation about an axis through the origin, and hence can be regarded as a rotation of the whole vector space R3. Thus each such rotation is represented by
an orthogonal 3x3 matrix whose i-th column is the image of the i-th
standard basis vector. This image of course is + or - a standard basis
vector, so the matrices describing the action of O3 on
R3
are the 24 3x3 matrices which have exactly one non-zero entry, which must
be 1 or -1, in each row and column and have determinant 1:
The other 24 matrices representing the indirect symmetries of the cube are obtained from these by mutltiplying exactly one entry by -1. What we have here then is a group of 24 (or 48) orthogonal transformations of R3 which is a faithful linear representation of O3. The 24 matrices representing O3 have determinant 1, the others determinant -1. Note that matrices in R3 with |det|=1 preserve areas, those with det=1 preserve orientations of triangles, and orthogonal matrices preserve lengths and angles. In general, let G be a linear group of finite degree over a field F. Denote the multiplicative group of non-zero elements of F by F*. Define a map det: G ->F* by M |-> det(M). Then det is a homomorphism whose kernel = {M in GL(n,F): det(M)=1} is a normal subgroup of GL(n,F) called the special linear group, denoted SL(n,F).
|
. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Go to Table of Contents.
To continue, click here. To return to Page 1, click here .
Last update Nov 25 1999
Author: Phill Schultz, schultz@maths.uwa.edu.au