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 $f : V \to W$,

$$\ker(f) \cong \operatorname{null}(A),$$

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?

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"Was that just an abuse of notation or have I had things mixed up all along?" Neither. Different courses/books will maintain/not maintain such a distinction. If a matrix represents some underlying linear transformation of a vector space, then the kernel of the matrix might mean the set of vectors sent to 0 by that transformation, or the set of lists of numbers (interpreted as vectors in $\mathbb{R}^n$ representing those vectors in a given basis, etc.