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linear algebra abstract vector spaces

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