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In this paper, we investigate the underlying fundamental relation between switched controllability and switched stabilization for switched linear control systems. It is shown by a counterexample that switched controllability does not imply switched stabilization by means of piecewise linear state feedback controllers. In addition, piecewise linear state feedback stabilizable and controllable switched systems may not be quadratic stabilizable. To illustrate, we take third-order systems as a case study and identify conditions for quadratic stabilizability and non-quadratic stabilizability.