Difference between a linear problem and a non-linear problem? Essence of Dot-Product and Kernel trick

Question

The kernel trick maps a non-linear problem into a linear problem.

My questions are:
1. What is the main difference between a linear and a non-linear problem? What i

Answer 1

The main difference (for practical purposes) is: A linear problem either does have a solution (and then it's easily found), or you get a definite answer that there is no solution at all. You do know this much, before you even know the problem at all. As long as it's linear, you'll get an answer; quickly.

The intuition beheind this is the fact that if you have two straight lines in some space, it's pretty easy to see whether they intersect or not, and if they do, it's easy to know where.

If the problem is not linear -- well, it can be anything, and you know just about nothing.

The dot product of two vectors just means the following: The sum of the products of the corresponding elements. So if your problem is

c1 * x1 + c2 * x2 + c3 * x3 = 0

(where you usually know the coefficients c, and you're looking for the variables x), the left hand side is the dot product of the vectors (c1,c2,c3) and (x1,x2,x3).

The above equation is (pretty much) the very defintion of a linear problem, so there's your connection between the dot product and linear problems.