Checking if something is a vector space Let $C^2[0,1]$ be the set of all fucntions f such that $f'$ and $f''$ are continuous on $[0,1]$
Now we have to determine if 
$w = {\{f \in C^2[0,1] : f'' + 4f = 0\}}$
is a vector space with the regular definitions of scalar multiplication and vector addition.
First I checked if $0$ is in $w$, which it is. 
But I am not sure how to determine if $w$ is a vector space. I think I must show that:
$a(f_1'' + 4f_1) + b(f_2''+4f_2) = 0$ with $f_1,f_2 \in w$. And this is obviously true as $0+0 = 0$ 
However I am pretty sure that this is not right, it seems a bit trivial. 
Thanks.

If you know about linear transformations, then this is clear because the set in question is the kernel of the linear operator $f \mapsto  f'' + 4f$.
