Kernel

We will first post a formal answer using technical notation, then we’ll give the answer with a technical sentence, then we’ll give a crude informal answer.

The kernel of a function : A B is the set[/latex]

The kernel of a function is the inverse image of the zero element.

Crude Informal: The kernel of a something is the set of elements that can be used (usually as arguments) with the something to make an answer of zero. The zero isn’t necessarily a number, it could be a zero vector, a xero matrix, etc.

Examples:

It is possible to take a nonzero matrix and multiply it against nonzero vectors and get a zero vector. The kernel of a matrix is the set of all vectors that the matrix can be multiplied against to result in a zero vector.

Appendix A:

It might be easier to get to the idea if we first consider the kernel for a function that takes numbers and gives numbers:

Ker (f) = { x : x A such that f(x) = 0}

The kernel of the function is all the elements from the original set that are mapped to zero by the function.