Given a linear map $L_t: V \to W$ represented by the matrix:
$ \begin{bmatrix} t & 0 & 1 & -1 \\ 0 & 2 & 1 & 0 \\ 1 & -2t & 0 & 1 \\ \end{bmatrix} $, consider $t \in \mathbb{R}$, when $L_t$ is injective and when it is surjective?
It is injective for the values of $t$ s.t. the restriction $L_{t_{\mathbb{R^3}}}$ is s.t. $dim(Im(L_{t_{\mathbb{R^3}}}))=3=\mathbb{R^3}$ and $dim(Ker(L_{t_{\mathbb{R^3}}}))=\{0\}$. $L_t$ cannot be surjective because $dim(W)>dim(V)$. Is that right?