I've been posed the following question:
I am unsure where to begin showing that $2F – G$ is still a linear transformation. Should I show that $2F-G$ conforms to the 3 properties of linear transformations?
Any suggestions/hints would be appreciated
Best Answer
You basically answered your own question; if you want to show that something is a linear transformation, just use the definition!
What will save the day (of course) is the hypothesis that both $F$ and $G$ are already linear transformations. You said three properties, but technically speaking you only need two, as respecting scalar multiplication will give you that $0$ is mapped to $0$.
So now just write it all out: $$ H(u+v)=2F(u+v)-G(u+v) = 2F(u)+2F(v)-G(u)-G(v) = \cdots $$ $$ H(cv)=2F(cv)-G(cv)=2cF(v)-cG(v) = \cdots $$