I'm taking basic linear algebra now. During a lecture on orthogonal complements, a professor said that there's a thing that is more 'general' than orthogonal complements and it's called the annihilator. He didn't provide further details, so I hopped on Google and found some explanations of this concept. Now I think I understand what the annihilator is, but I can't seem to prove a basic fact about it..
Theorem 3.14, part (1) in Advanced Linear Algebra by Steven Roman (Google Books) says that
$M \subseteq N \Rightarrow N^0 \subseteq M^0$
I don't know how to prove this (the author leaves proof of it for the reader). Intuitively I understand why $M^0$ can't be smaller than $N^0$, for if we extend the subset $M$ by adding some $n \in N$ to it, then, in case (a), some functionals from the 'original' $M^0$ won't be able to send the new element of $M$ to zero and $M^0$ will get smaller. In case (b), every functional from the 'original' $M^0$ will send $n$ to zero, and $M^0$ will stay the same. But $M^0$ can't get bigger, because the functionals that annihilate $n$ AND all the vectors in $M$ must be in $M^0$ already.
So can you please help me convert my vision of the problem into a formal proof?