This is a question from my lecture notes, but I was unable to follow what my teacher said. The exact question is:
Is the following subset of subspace in $C[a,b]$, the space of all continuous real-valued function on $[a,b]$? Let $W$ be the set of all differentiable functions such that $f'(x) = xf(x)$. I need to determine whether $W$ represents a vector subspace.
Best Answer
The zero function is a constnat function so it's derivative is zero, therefore it holds
$f'(x)=x.f(x)$ $\forall x$ $\Rightarrow$ $0\in W$. Let $f,g \in W \\$.
We have $(f+g)'(x)=f'(x)+g'(x)=x.f(x)+x.g(x)=x.(f+g)(x)\\$
Therefore $(f+g)\in W.\\$ Let $f\in W$ and $a=const$.
Then $(af)'(x)=a.f'(x)=a.x.f(x)=x(af)(x)\Rightarrow af\in W.\\$
So it follows that $W$ is a vector space.