Finding matrix for parallel transport map. Consider the surface $S$ given by the patch $$\sigma (u, v) = (u, v, 0)$$ and the points $$p(0, 0, 0),\ q(1, 1, 0) \in S.$$ Choose bases for $T_pS$ and $T_qS$ and write down the matrix for the parallel transport map $P_{\gamma_{p,q}}$ where $\gamma(t) = (t, t, 0)$.

The surface that you have is just a flat plane. In that case, parallel transport is just good, old-fashioned translation, i.e. to parallel transport a vector you slide it around the plane while keeping it parallel to,  and pointing in the same direction as, the original. 
Hence, the matrix would be the two-by-two identity matrix.
The parallel transport matrices, also called the holonomy group, give you an idea of the curvature of a space. Because the plane is simply connected and, in a specific sense, "flat"; the parallel transport matrices are the identity matrices and the holonomy group is the trivial group.
