Inspired by prior work by Tian and by Cao and Xu, this paper presents an efficient computer-aided framework to characterize the fundamental limits of coded caching systems under the constraint of linear coding. The proposed framework considers non-Shannon-type inequalities which are valid for representable polymatroids (and hence for linear codes), and leverages symmetric structure and problem-specific constraints of coded caching to reduce the complexity of the linear program. The derived converse bounds are tighter compared to previous known analytic methods, and prove the optimality of some achievable memory-load tradeoff points under the constraint of linear coding placement and delivery. These results seem to indicate that small, structured demand subsets combined with minimal common information constructions may be sufficient to characterize optimal tradeoffs under linear coding.