Non-linear SVM classifier is supposed to give better accuracy over linear. Is your linear SVM far worse than the non-linear SVM/Kernel? May be delta accuracy is not large enough to discard the linear classifier.
There is a nice discussion here (https://stats.stackexchange.com/questions/73032/linear-kernel-and-non-linear-kernel-for-support-vect...)
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Use this tutorial as a handy guide to weigh the pros and cons of these commonly used machine learning algorithms.
Find more tutorials on the SAS Users YouTube channel.