# Run-Time difference between linear and RBF kernels in SVM

I'm using kernlab package in R. I trained an SVM using linear kernel and RBF kernel with same data set (the number of instances is 3000). When I use a linear kernel, it takes much longer to train the model compared to when I use an RBF kernel (accuracies are similar). Why does this run-time difference exist?

First, you need to compute the kernel matrix (i.e. the matrix populated with $K(x_i,x_j)$). Normally, this should take longer for the RBF kernel than for the linear kernel (the linear kernel does not have to call $\exp$ which is the most expansive part).
However, the time of the training part is also a significant part of the training process which consists of solving the quadratic program is variable in $C$ the cost parameters and the kernel matrix of the training samples.