I'm being asked to consider $T: \mathbb{R}^3 \to \mathbb{R}^3$ given by the formula: $$(u, v, w) = T(x, y, z) = (x-y, y-z, z-x)$$
Then, I'm asked under what conditions, if any, does a vector $(u, v, w)$ lie in the image of $T$?
If my understanding is correct, the term "image" refers to the range of a transformation, meaning any element mapped to by $T$ within the codomain is the image.
It looks to me that any value for $u$, $v$, or $w$ will be acceptable under $T$, but I don't know how to validate that. I think I need some sort of equation that will prove that any input will produce a valid output. Is this a valid line of thinking, or am I on the wrong track?
My question is: How do I prove that any value is acceptable for a valid output? Thank you.
$(u,v,w)$ will lie in the image of $T$ precisely if it is a linear combination of the columns of $T's$ matrix. Namely, $(u,v,w)\in\rm{span}\{(1,-1,0),(0,1,-1), (-1,0,1)\}=\rm{span}\{(1,-1,0),(0,1,-1)\}$.
That is, the vector must lie on the plane $x+y+z=0$.