# What is the difference between kernel and null space?

What is the difference, if any, between kernel and null space?

I previously understood the kernel to be of a linear map and the null space to be of a matrix: i.e., for any linear map $f : V \to W$,

where

• $\cong$ represents isomorphism with respect to $+$ and $\cdot$, and
• $A$ is the matrix of $f$ with respect to some source and target bases.

However, I took a class with a professor last year who used $\ker$ on matrices. Was that just an abuse of notation or have I had things mixed up all along?