What you have done is perfectly correct.
I'm not sure how to type the problems out using this forum so I attached them below.
For 1.1
I have:
Since u,v are in V L[u]=0, L[v] = 0
L[c1u + c2v] = L[c1u] + L[c2v]
by the rules of differentiation, the constant can also be pulled out in front
so, L[c1u] + L[c2v] = c1*L[u] + c2*L[v] and since L[u]=0, L[v] = 0
the whole thing = 0.
Does this prove 1.1? Or do i need more than this?
For 1.3
I have:
i need to show that H is closed under vector addition and scaling.
Let c be a scalar, c is in R and f,g are in c[a,b]
(f+g)(a) = f(a) + g(a) = f(b) + g(b) = (f+g)(b)
(cf)(a) = c[f(a)] = c[f(b)] = (cf)(b)
therefore, H is a subspace.
Is this a correct way of proving 1.3?
Can someone please check these for me, I really need to understand them. Thank you