If I understand correctly, this question is asking for a 3x2 matrix $A$, where $x$ is the set of vectors $(x,y,z)$ that satisfy the equation $x - y + 2z = 0$, or $Ax = 0$. I am having difficulty understanding how to set this up. If I am misunderstanding the question, I would appreciate any help!
No such $T$ exists. By condition (b), $\operatorname{rank}T=2$, so by the rank–nullity theorem $\dim\ker T=0$, i.e. $T$'s nullity is 0 (the domain is of dimension 2). This means that $T$ is injective (one-to-one), but this contradicts condition (a).
To find a $T$ satisfying just (b), note that $x-y+2z=0$ is a plane in $\mathbb R^3$ whose normal vector is $(1,-1,2)$. This plane can be parametrised by two (linearly independent) vectors orthogonal to $(1,-1,2)$, say $(1,1,0)$ and $(-1,1,1)$. The desired $T$ is then those two new vectors combined into a matrix: $$T=\begin{bmatrix}1&-1\\1&1\\0&1\end{bmatrix}$$ Left-multiplying a vector in $\mathbb R^2$ with this $T$ is simply translating the parameters (contained in the $\mathbb R^2$ vector) into a point in $\mathbb R^3$.