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

1. Linear equations are homogeneous, and superposition applies. You can create solutions using combinations of other known solutions; this is one reason why Fourier transforms work so well. Non-linear equations are not homogeneous, and superposition does not apply. Non-linear equations usually have to be solved numerically using iterative, incremental techniques.
2. I'm not sure how to express the importance of the dot product, but it does take two vectors and returns a scalar. Certainly a solution to a scalar equation is less work than solving a vector or higher-order tensor equation, simply because there are fewer components to deal with.

My intuition in this matter is based more on physics, so I'm having a hard time translating to AI.