A linear algebraic group is a matrix group that is also an affine variety. In particular, its elements satisfy polynomial equations. The group operations are required to be given by regular functions. The linear algebraic groups are similar to the Lie groups, except that linear algebraic groups may be defined over any field, including those of positive field characteristic.
The special linear group of matrices of determinant one 
is a linear algebraic group. This is because the equation
for the determinant is a polynomial
equation in the entries of the matrices. The general
linear group of matrices with nonzero determinant 
is also a linear algebraic group. This can be seen by
introducing an extra variable 
and writing
 |
This is a polynomial equation in 
variables and is equivalent
to saying that 
is nonzero. This equation
describes 
as an affine variety.
