Does it matters that hyerplane is linear or non-linear? I think with hyperplane you mean the actual decision boundary of the SVM. The decision boundary is always linear, at least in the transformed feature space. In pictures you often see a wiggling line which separates the datapoints (the decision boundary, looks not linear). But in fact, the actual classification took place in the much higher dimensional feature space, where the decision boundary can be seen as linear.

Yash Patel: A SVM is in principle a linear classifier. In easy words, a SVM tries to separate datapoints in a linear way, hence with a linear decision function. It is often shown as a line trough datapoints. But often data is not separable by a simple line. Therefore we use Kernels. We expand the feature space, for example 2dimensional (feature1, feature2) to a much higher feature space. This transition from lower feature space to higher feature space can be calculated with more or less no costs with the Kernel Trick (inner products).

user20160: this is right, I was to sloppy here.