Show that composition by $\varphi$ is a linear mapping 
Let $\varphi$ be any mapping from a set $A$ to a set $B$. Show that composition by $\varphi$ is a linear mapping from $\mathbb{R}^B$ to $\mathbb{R}^A$. That is, show that $T:\mathbb{R}^B \rightarrow \mathbb{R}^A$ defined by $T(f) = f \circ \varphi$ is linear.


To show that a mapping is linear, we must demonstrate that the vector operations are preserved. 
I am unclear about the wording here "composition by $\varphi$ is a linear mapping from $\mathbb{R}^B$ to $\mathbb{R}^A$" which implies to me that the composition by $\varphi$ leads to an inverse of $\varphi$ mapping defined previously from set $A$ to set $B$, but I don't know why. I am getting a lot of confusion here and I think that I am interpreting the problem wrong.
I believe I have to show that $$f(\varphi_1 + \varphi_2) = f(\varphi_1) + f(\varphi_2)$$ and $$cf(\varphi) = f(c \varphi)$$ to show that the mapping is linear, but I am not sure how to get started. 

$T$ is a function on $\mathbb{R}^B$, so you need to show it is linear on this space. That is, if $c\in\mathbb{R},$ and $f,g\in \mathbb{R}^B,$ then you need to show that $$T(cf)=cT(f),$$ and $$T(f+g)=T(f)+T(g).$$ You just need to use composition properties to show these; I'm sure you can take it from here.
