Must every right-inverse of a linear transformation be a linear transformation? Let T be a linear transformation $\mathbb{R}^2 \rightarrow \mathbb{R}^3$. Let S be the right-inverse of T. Does S have to be linear transformation?
Thanks in Advance.

No
(Actually a linear map $\mathbb R^2 \to \mathbb R^3$ cannot have a right inverse, since having a right inverse is equivalent to being surjective, and linear maps have the dimension of the range at most the dimension of the domain, so there are no surjective linear maps $\mathbb R^2 \to \mathbb R^3$)
BUT ANYWAY. . . .
Consider the projection map $P: \mathbb R^2 \to \mathbb R$ given by $P(x,y) = x$. Let $f : \mathbb R \to \mathbb R$ be your favourite nonlinear function and define the right-inverse $Q: \mathbb R \to \mathbb R^2$ by $Q(x) = (x,f(x))$. 
Then we have $P\circ Q(x) = P(Q(x))=P(x,f(x))= x$ so this is indeed a right inverse.
To see $Q$ is nonlinear observe the image is the graph of the function $f$ which is not a linear subspace of $\mathbb R^2$. That means $Q$ is nonlinear.
