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. 

The kernel of a linear map ($T$) between two vector spaces $U$ and $V$ are those elements ($v$) of $V$ such that $T(v)=0$. It is defined analogously for other structures.
You can see it as a measure of how far is the map from being one to one.
