<p>The 2-dimensional Toda lattice (2D Toda) is a completely integrable semi-discrete wave equation with the KP-II equation in its continuous limit. Using Darboux transformations, we prove the linear stability of 1-line solitons for 2D Toda of any size in an exponentially weighted space. We prove that the dominant part of solutions for the linearized equation around a 1-line soliton is a time derivative of the 1-line soliton multiplied by a function of time and transverse variables. The amplitude is described by a 1-dimensional damped wave equation in the transverse variable, as is the case with the linearized KP-II equation.</p>