I've been given the following problem as homework:
Prove that there exists a linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^3$ such that $T(1,1) = (1,0,2)$ and $T(2,3) = (1,-1,4)$.
Since it just says prove that one exists, I'm guessing I'm not supposed to actually identify the transformation.
One thing I tried is showing that it holds under addition/multiplication in the sense of:
1) $T(x + y) = T(x) + T(y)$
2) $T(cx) = cT(x)$
3) $T((1,1) + (2,3)) = T(1,1) + T(2,3)$? But that's not necessarily true.
I don't know how I'd accomplish this given my limited knowledge.
What's the approach to solving a problem like this?