Kernel in Modern Algebra What is a Kernel and how can it be describe in the real world and how can it be defined well and precisely. Tried asking my professor and he just tell us it is just abstract idea. 

One area in which kernels of linear operators have a real-world physical meaning is electrical networks. There, you have a network consisting of set of nodes and a set of branches connecting these nodes, together with a “boundary operator,” $\partial$, that basically tells you which nodes are at the ends of each branch and assigns a direction to each branch. You can define a vector space indexed by the branches that describes assignments of currents to the branches and a related vector space indexed by the nodes that gives the net current flowing into or out of each node. The boundary operator then tells you how to compute the net node currents given a set of branch currents. The kernel of $\partial$ consists of those branch current assignments for which the net current at all nodes is zero, but that’s exactly Kirchhoff’s current law for electrical networks, so the kernel of $\partial$ captures all of the physically possible current distributions. There are other operators that can be defined on these spaces and their duals whose kernels and images also have important physical meanings. 
