The increase in anthropogenic CO2 emissions largely followed an exponential path between 1850 and 2010, and the corresponding increases in atmospheric CO2 concentration were almost constantly proportional to the emissions by the so-called "airborne fraction". These observations suggest that the dynamics of atmospheric CO2 concentration through this time period may be properly approximated as a linear system. We demonstrate this hypothesis by deriving a linear box-model to describe carbon exchanges between the atmosphere and the surface reservoirs under the influence of disturbances such as anthropogenic CO2 emissions and global temperature changes. We show that the box model accurately simulates the observed atmospheric CO2 concentrations and growth rates across interannual to multi-decadal time scales. The model also allows us to analytically examine the dynamics of such changes/variations, linking its characteristic disturbance-response functions to bio-geophysically meaningful parameters. In particular, our results suggest that the elevated atmospheric CO2 concentrations have significantly promoted the gross carbon uptake by the terrestrial biosphere. However, such "fertilization" effects are partially offset by enhanced carbon release from surface reservoirs promoted by warmer temperatures. The result of these interactions appears to be a decline in net efficiency in sequestering atmospheric CO2 by ∼30% since 1960s. We believe that the linear modeling framework outlined in this paper provides a convenient tool to diagnose the observed atmospheric CO2 dynamics and monitor their future changes.