Are "eigenspace" and "null space" ever *not* synonymous?

Question

Both the null space and the eigenspace are defined to be "the set of all eigenvectors and the zero vector". They have the same definition and are thus the same. Is there ever a scenario where the null space is not the same as the eigenspace (i.e., there is at least one vector in one but not in the other)?

Answer 1

Let $A$ denote a linear operator from a vector space $V$ to $V$, e.g. a square matrix. One definition of an eigenspace is a set of the form $\{v\in V|Av=\lambda v\}$. (I prefer this to your version because it makes the zero vector sound like a special fudge.) As I understand it, "null space" refers not to an arbitrary eigenspace, but to the $\lambda=0$ special case. Of course, that's also called the kernel of $A$ (denoted $\operatorname{ker}A$), so either way "null space" is synonymous with some other term.

Answer 2

Only linear maps $L:V\to V$ from a vector space to itself can have eigenvectors, eigenspaces, and so on. Linear maps $M:V\to W$ between different vector spaces have null spaces, but cannot have eigenspaces.