Thread: 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

What you have done is perfectly correct.

Oh for the 2nd one do I also need to show that the zero vector is in H? I forgot about that

Originally Posted by SpiffyEh

Oh for the 2nd one do I also need to show that the zero vector is in H? I forgot about that

Yes, a subspace must be nonempty! Same for the first one also.

How do i show that those two subspaces are nonempty?

Originally Posted by SpiffyEh

How do i show that those two subspaces are nonempty?

Show that they contain the zero vector!

I understand that I just don't see how to show that they do contain a zero vector. I think for the 2nd one if I use a constant = 0 that would show it correct? But for the first I'm not sure unless its the same idea.

Originally Posted by SpiffyEh

I understand that I just don't see how to show that they do contain a zero vector. I think for the 2nd one if I use a constant = 0 that would show it correct? But for the first I'm not sure unless its the same idea.

Of course! That would be correct! For the first one, just choose the zero function - it certainly satisfies any homogeneous linear differential equation.

Oh ok, I can do that Thank you