Linear Transformations · Challenge problem

A linear map $T:\mathbb{R}^2\to\mathbb{R}^2$ sends every vector on the line $y = x$ to a vector on the line $y = x$, and every vector on the line $y = -x$ to a vector on the line $y = -x$. If $T(1,0) = (2,1)$, find the first coordinate of $T(0,1)$.