$B = \left(\begin{matrix} 0.4 & 0 & 0 & 0 \\ 0 & 0.4 & 0 & 0 \\ 0 & 0 & 0.4 & 0 \end{matrix}\right)$
Here, $B \in \mathbb{C}^{3 \times 4}$ where $\mathbb{C}$ is complex field. Let $T = \mathbb{C}^3$ and $e = (0,0,0,1)^t \in \mathbb{C}^4$ and $S = \mathrm{span}(e)$. $T$ and $S$ are vector spaces.
How to show that $\mathrm{range}(B) \subset T$ and can we conclude the similar relation between the null space of $B$ and $S$?