Recently, Reddy (2018) has studied a posteriori parameter choice rules for the weighted Tikhonov regularization scheme. In this article, we consider the finite dimensional weighted Tikhonov scheme and discuss the convergence analysis of the scheme. We introduce a class of parameter choice strategies to choose the regularization parameter in the finite dimensional weighted Tikhonov scheme; and derive the optimal rate of convergence O(δα+1α+2) for the scheme based on the proposed strategies. Results of numerical experiments are documented to illustrate the performance of the scheme with an efficient finite dimensional approximation of an operator.