SVM is a linear classifier. But some articles talk about non-linear SVM that is quite contradictory. A "non-linear SVM" can perform non-linear classification over a dataset that is not linearly separable. What we do is increase the dimension of the dataset to make it linearly separable and present it to SVM. So the SVM remained linear and we format the dataset to feed it to the SVM classifier. Is my concept wrong somewhere?
Wikipedia explains nonlinear SVM classification with
The original maximum-margin hyperplane algorithm proposed by Vapnik in 1963 constructed a linear classifier. However, in 1992, Bernhard E. Boser, Isabelle M. Guyon and Vladimir N. Vapnik suggested a way to create nonlinear classifiers by applying the kernel trick (originally proposed by Aizerman et al.) to maximum-margin hyperplanes.
The resulting algorithm is formally similar, except that every dot product is replaced by a nonlinear kernel function. This allows the algorithm to fit the maximum-margin hyperplane in a transformed feature space.
The transformation may be nonlinear and the transformed space high-dimensional; although the classifier is a hyperplane in the transformed feature space, it may be nonlinear in the original input space.