  In this paper we construct a faithful representation of the mapping class
group of the genus two surface into a group of matrices over the complex
numbers. Our starting point is the Lawrence-Krammer representation of the braid
group B_n, which was shown to be faithful by Bigelow and Krammer. We obtain a
faithful representation of the mapping class group of the n-punctured sphere by
using the close relationship between this group and B_{n-1}. We then extend
this to a faithful representation of the mapping class group of the genus two
surface, using Birman and Hilden's result that this group is a Z_2 central
extension of the mapping class group of the 6-punctured sphere. The resulting
representation has dimension sixty-four and will be described explicitly. In
closing we will remark on subgroups of mapping class groups which can be shown
to be linear using similar techniques.
