S. Bigelow proved that the braid groups are linear. That is, there is a faithful representation of the braid group into the general linear group of some field. Using this, we deduce from previously known results that the mapping class group of a sphere with punctures and hyperelliptic mapping class groups are linear. In particular, the mapping class group of a closed orientable surface of genus $2$ is linear.
KORKMAZ, MUSTAFA (2000) "On the Linearity of Certain Mapping Class Groups," Turkish Journal of Mathematics: Vol. 24: No. 4, Article 5. Available at: https://journals.tubitak.gov.tr/math/vol24/iss4/5