Nullity of a linear transformation $T$ > T is a projection onto the vector $v = (1, 2, 2)$ given by $$T(x,y,z) = \frac{x+2y+2z}{9} \cdot (1, 2, 2)$$ Use the given information to find the nullity of $T$. I know the nullity of $T$ is determined by $$x+2y+2z= 0$$ Is there a systematic way of finding the nullity of $T$?

A single non-trivial linear relation in a three-dimensional space determines a plane. Thought differently, there are three variables and one constraint, so there are two free variables. And so the nullity is $2$.

Implicitly, I'm thinking of the rank-nullity theorem.