So I know that the set of vectors $Ax = 0$ is called the null space or the kernel of A; and I know that we call the set of vectors b such that $Ax = b$ does have a solution the range or the column space of A.
But, what do we call the set of vectors b such that $Ax = b$ has no solution? Do we have such a name? I feel like there's a connection between the set of vectors b that have no x solution and the null space.
Assuming you have $T_A:X\to Y$ via $T_A\mathbf{x} = A\mathbf{x}$, the set $Y\setminus \operatorname{im} T_A$ is what you describe. But it doesn't have a name that I'm aware of. It isn't a vector space (because it doesn't contain $\mathbf 0$). And it totally depends on what you regard as the larger space $Y$ containing $\operatorname{im} T_A$ is, so it isn't natural (not being determined by $A$) and probably not very interesting either.