Difference between kernel of a matrix and kernel of a transformation?

Question

Does the word "kernel" carry different meanings in the contexts of matrices and transformations? Or are the definitions of the two in those two contexts intertwined in some way?

I know the kernel of a matrix to be it's null space. I haven't been taught about the kernel of a transformation yet, but I looked it up and it's a bunch of complicated stuff that doesn't make much sense (it's apparently every vector v that results in a transformation T outputting the zero vector?)

Are they just two different ideas?

Answer 1

The kernel of a matrix $A$ is the space of those vectors $v$ such that $A.v=0$. The kernel of a linear transformation $T$ is the space of those vectors $v$ such that $T(v)=0$. So, it is basically the same thing.

Answer 2

Every linear transformation from one finite dimensional space to another can be written as a matrix. The only difference between "kernel of a linear transformation" and "kernel of a matrix" would be in the case of a linear transformation over infinite dimensional spaces which cannot be written as a matrix. An example would be the "differentiation" operator on the space of all differentiable functions. The kernel of that operator is the subspace of all constant functions.