Answer
H is a subspace of $M_{2\times 2}$.
Work Step by Step
See definition on p.162.
(a) The zero matrix has zero in the lower left corner
0$\in$ H.
(b) The sum of two matrices from H will have 0+0=0 in the lower left corner. H is closed under addition.
(c) If A$\in$H, it has a zero entry in the lower left corner.
cA has c(0)=0 in the lower left corner. cA$\in$H
H is closed under scalar multiplication.
Conclusion:
By definition of subspace, H is a subspace of $M_{2\times 2}$.
