Kernel functions are the functions that, given the original feature vectors, return the same value as the dot product of its corresponding mapped feature vectors. Kernel functions do not explicitly map the feature vectors to a higher-dimensional space, or calculate the dot product of the mapped vectors. Kernels produce the same value through a different series of operations that can often be computed more efficiently.

The main reason for using kernel functions is to eliminate the computational requirement to derive the higher-dimensional vector space from the given basic vector space, so that observations be separated linearly in higher dimensions. Why someone needs to like this is, derived vector space will grow exponentially with the increase in dimensions and it will become almost too difficult to continue computation, even when you have a variable size of 30 or so. The following example shows how the size of the variables grows.

Example: When we have two variables such...