# "Fixed Domains" of a Linear Transformation

Given a linear transformation $T$, I need to find the set of all domains $D$ such that $T:D\mapsto D$. Equivalently, I need to find the set of all domains $D$ that are symmetric under $T$.

Aside from solving this particular question, what sort of theory should I study to solve this sort of question?

• By domain, you just mean subset, right? Do you require $T(D) = D$ or is $T(D) \subset D$ good enough? Commented Dec 20, 2013 at 13:41
• No, I need equality. In context, I need to find the set of all functions $f:D \mapsto \mathbb{R}$ s.t. $Tf=f$, or ideally prove that no such functions exist (in the context in which I'm working), and I thought a good first step would be identifying the domain $D$ on which such a function must be defined. Commented Dec 20, 2013 at 14:03
• Wait. You seem to be describing two different questions in the original question vs. your comment. Commented Dec 21, 2013 at 4:42
• Right. First, in retrospect even if I had the set of all $f$ the first thing I'd do is find the union of their domains anyway; second, like I said, I need the answer to this as an intermediate result and it's what my actual question is for now. I might post the original as a separate question if I have trouble making the jump. Commented Dec 21, 2013 at 4:52
• The question remains unclear. On what space does the transformation $T$ act? Commented Dec 25, 2013 at 3:40