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.

Demonstrating that $0 \in w$ tells us that the vector space is nonempty.
If $f_1$ and $f_2$ are in $w$ then $f_1'' + 4f_1 = 0$ and $f_2''+4f_2 = 0$.
If we take $h=af_1 + bf_2$, then $$h''+4h = (af_1''+bf_2'')+4(af_1+bf_2) = a(f_1''+4f_1) + b(f_2''+4f_2)=0+0 =0.$$
Thus we see that a linear combination of the functions $f_1$ and $f_2$ are again in $w$, and $w$ is a vector space.
