T is a linear transform/map or (it is the same thing!) a homomorphism, or a matrix: an application between 2 vector spaces that preserves vector addition and sc

T is a linear transform/map or (it is the same thing!) a homomorphism, or a matrix: an application between 2 vector spaces that preserves vector addition and scalar multiplication (i.e. f(x+y) = f(x)+f(y), f(a.x) = a.f(x)). T : V -> W is the signature of this homomorphism, whereas ker(T) = {v \in V | T.v = 0} means "ker of T is the set of vectors v in the vector space V such that T applied to v is the neutral element of W". The neutral element is noted 0 because, in a vector space, it is such that v + neutral_element = v. So neutral_element is the null vector. I learned algebra kernels in my first year of college, so I assumed wrongly. But that's not the point: I was not thinking about explaining what a kernel is but instead (higher level) trying to group what seemed to be scattered definitions into explaining why we call "kernel machines" this (and why a kernel trick involves a kernel). Now, I'm thinking about explaining (with pictures) what an algebra kernel is...