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