Ok, I know that the title doesn't make much sense, but I didn't know how to describe this question, and I didn't want to simply put "linear algebra question". Anyway:
We have $f: \mathbb R^3 → \mathbb R^3$ defined by
$f(x,y,z)=(x−y+2z,−x+y−z,2x−y+z)$
If I have $(a,b,c)\in \mathbb R^3$, under what conditions do we have that $(a,b,c)\in f(\mathbb R^3)$.
I'm not really sure how to go about this problem. Would the conditions be the same as the conditions I found when proving that this function is surjective? (Proving that a multivariable function is surjective?) My intuition tells me that the conditions are the same, but I'm still not sure as to how this is supposed to be answered formally. Any guidance is appreciated, thanks.