In this paper, a new adaptive control framework for linear systems in which the matched uncertainty can be linearly parameterized is introduced to guarantee the global exponential stability of reference tracking error and parameter convergence error without requiring restrictive persistent excitation condition. The framework uses time histories of control input and system signals to construct least-squares problem based on recorded data. Then, unique solution to least-squares problem is computed, and assigned as pre-selected value in well-known sigma-modification term. Such indirect use of recorded data matrices results in globally exponential convergence of tracking error and parameter convergence error provided that the recorded matrix satisfies the simple rank condition. The proofs are given by Lyapunov stability theorem, and the results are illustrated with simulations.