Given that the linear transformation $\ltdefn{T}{\complex{3}}{\complex{5}}$, $\lt{T}{\colvector{x\\y\\z}} = \begin{bmatrix} 1 & 3 & 2\\ 0 & 1 & 1\\ 1 & 2 & 1\\ 1 & 0 & 1\\ 3 & 1 & 2 \end{bmatrix} \colvector{x\\y\\z}$ 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.