Intuitively, if a linear system is controllable, we presumably can excite all the system modes (or states) from the input. The controllablility matrix (or the range of the controllability matrix) tells us which states are controllable, so if this range is the entire state space, we should be able to excite any or all modes we wish. A more mathematical answer has to do with the form of the particular solution the system ODEs, and since I'm limited to HTML coding here, I'm not going to try to write down any complicated mathematics....if you're interested in more discussion, just send me an e-mail (jshumber@cds.caltech.edu) to make an appointment or come to the review session this Friday.