Recently, Nair and Roy (2017) considered a linear regularization method for a parameter identifica-tion problem in an elliptic PDE. In this paper, we consider similar procedure for identifying the diffusion coef-ficient in the heat equation, modifying the Sobolev spaces involved appropriately. We derive error estimatesunder appropriate conditions and also consider the finite-dimensional realization of the method, which isessential for practical application. In the analysis of finite-dimensional realization, we give a procedure toobtain finite-dimensional subspaces of an infinite-dimensional Hilbert spaceL2(0,T;H1(Ω))by doing dou-ble discretization, that is, discretization corresponding to both the space and time domain. Also, we analyzethe parameter choice strategy and obtain an a posteriori parameter which is order optimal.