If $\lt{T}{\colvector{x\\y\\z}} = \zerovector$, then $\colvector{2x + y + z\\x - y + 2z\\x + 2y - z} = \zerovector$. Thus, we have the system
\begin{align*}
2x + y + z &= 0\\
x - y + 2z &= 0\\
x + 2y - z &= 0
\end{align*}
.
Thus, we are looking for the nullspace of the matrix
$A_T =
\begin{bmatrix}
2& 1 & 1\\
1 & -1 & 2\\
1 & 2 & -1
\end{bmatrix}$.
Since $A_T$ row-reduces to
$\begin{bmatrix}
\leading{1} & 0 & 1\\
0 & \leading{1} & -1 \\
0 & 0 & 0
\end{bmatrix}$,
the kernel of $T$ is all vectors where $x = -z$ and $y = z$.
Thus, $\krn{T} = \spn{\set{\colvector{ -1\\1\\1}}}$.
Since the kernel is not trivial, Theorem KILT tells us that $T$ is not injective.