I've found that basis for W is $ \{(1,1,0,0,0)^T \ $, $(0,0,1,-2,1)^T \} $ and basis required in part (b) to be $ \{(-1,1,0,0,0)^T \ $, $(0,0,2,1,0)^T \ $, $(0,0,-1,0,1)^T \ $}. How can I verify the statement in part (c) and subsequently that in part (d)?
Note that
$$(0,0,1,1,1)=(0,0,2,1,0)+(0,0,-1,0,1),$$
$$(-1,1,1,2,3)=(-1,1,0,0,0)+2(0,0,2,1,0)+3(0,0,-1,0,1),$$ and
$$(0,0,1,3,5)=3(0,0,2,1,0)+5(0,0,-1,0,1).$$
Is this true for all matrices? Well, note that elements of $W$ are perpendicular to the row space. So $W^{\perp}$ must be the row space.