Given that the linear transformation $\ltdefn{T}{\complex{3}}{\complex{3}}$, $\lt{T}{\colvector{x\\y\\z}} = \colvector{2x + y\\2y + z\\x + 2z}$ is injective, show directly that $\set{ \lt{T}{\vect{e}_1},\, \lt{T}{\vect{e}_2},\, \lt{T}{\vect{e}_3} }$ is a linearly independent set.