how to map the data into high dimensional feature space using linear/poly kernel?

I have 2 questions: is mapping the data to feature space explecitly, is it same as using the kernel

here I transformed the data to feature space now should I use a kernel or how it actually done? and is this the right way?

def feature_map(X):
     return np.asarray(X[:, 0], X[:, 1], np.dot((X[:, 0]), X[:, 1]))) # is this feature map or kernel space 

can some one please explain how to think about feature mapping?