Linear Transformations · Superchallenge problem

Among all linear maps $T:\mathbb{R}^3\to\mathbb{R}^3$ with $T(1,1,0) = (1,2,0)$ and $T(1,0,1) = (0,1,1)$, find the smallest possible value of the sum of the squares of the nine entries of the standard matrix of $T$.