This thesis is about a linear algebraic object called a tridiagonal pair. Roughly speaking, a tridiagonal pair on a vector space V is a pair of diagonalizable linear transformations on V that each act on the eigenspaces of the other in a tridiagonal fashion. This thesis is divided into two parts. Below we give a brief overview of what each part is about. The first part of this thesis is about two commuting linear transformations associated with a tridiagonal pair. Given a tridiagonal pair on V, we introduce two linear transformations Δ :V → V and Ψ :V → V that we find attractive. We discuss how Δ, Ψ act on the first and second split decomposition of V. We describe Δ, Ψ from several points of view and show how they are related to each other. Along this line we have two main results. Our first main result is that Δ , Ψ commute. In the literature on tridiagonal pairs, there is a scalar β used to describe the eigenvalues. Our second main result is that each of Δ^{± 1} is a polynomial of degree d in Ψ, under a minor assumption on β. The second part of this thesis explores a connection between tridiagonal pairs and the quantum enveloping algebra Uq(sl2). In this part, we focus on tridiagonal pairs of q-Racah type. This is the most general type of tridiagonal pair. We define two linear transformations K:V → V and B:V → V which act on the split decompositions in an attractive way. Using Ψ, K,B we obtain two Uq(sl2)-module structures on V. For each of the Uq(sl2)-module structures, we compute the action of the Casimir element on V. We show that these two actions agree. Using this fact, we express Ψ as a rational function of K^{± 1}, B^{± 1} in several ways. Eliminating Ψ from these equations we find that K and B are related by a quadratic equation.