Differnece between the Kernel and the Nullspace?

HallsofIvy

As others have said, in Linear Algebra, "kernel" (of a linear transformation) and "nullspace" are the same thing. One can show that the kernel of any linear transformation, from one vector space to another, is subspace (perhaps trivial) of the domain space.

If u and v are in the nullspace of L, then L(u+ v)= L(u)+ L(v)= 0+ 0= 0 so the null space is closed under addition. If u is in the nullspace of L and k is any scalar, then L(ku)= kL(u)= 0 so the null space is also closed under scalar multiplication. Therefore it is a subspace and the name "null space" is justified. I suppose one could use the term "kernel" for sets, not subspaces, such that f(u)= 0 for some function f that is NOT a linear transformation, but the "linearity" is, after all, the whole point of vector spaces.

Given a function from one algebraic object to another, other than vector spaces, the term "kernel" is used to mean the set of points in one that are mapped into the additive identity of the other.