Answer
H is a subspace of $M_{2\times 4}$.
Work Step by Step
(a)
The zero matrix from $M_{2\times 4}$ is such that $F0=0$
0$\in$ H.
(b)
If A,B$\in$H, then FA=0 and FB=0.
F(A+B)=FA+FB$ \quad$ (left distributive law, Th.2b, sec 2-1)
F(A+B)=$0+0=0$, so A+B$\in$H.
H is closed over addition.
(c)
If A$\in$H, then FA=0
F(cA)=c(FA) $\quad$... (Th.2.d, sec 2-1)
F(cA)=$c0=0$, so
H is closed over scalar multiplication.
Conclusion:
By definition of subspace, H is a subspace of $M_{2\times 4}$.
H is a subspace of $M_{2\times 4}$.
