Summary: Domain is column vectors, codomain is matrices. Domain is dimension 3 and codomain is dimension 4. Not injective, not surjective.
Definition:\begin{align*}\ltdefn{T}{\complex{3}}{M_{22}},\quad \lt{T}{\colvector{a\\b\\c}}= \begin{bmatrix} a-b&2a+2b+c\\ 3a+b+c&-2a-6b-2c \end{bmatrix} \end{align*}