Let $S \left(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}\right) = $ $\begin{pmatrix} x_1 & -2x_2 & x_3 & x_4\\ 2x_1 & - 4x_2 & -3x_3 & -3x_4\\ x_1 & -2x_2 &-4x_3 & -3x_4 \end{pmatrix} $
Determine if this linear transformation is surjective or injective. I already determined that it is not injective because the $\dim(Ker(S)) \not= 0$ therefore it is not injective. And I know a transformation is surjective iff $\dim(Range(S)) = $ dimesion of the codmain. However in a question like this the codomain is not given so its kind of throwing me off. Any help would be great!