The set of differentiable real-valued functions $f$ on the interval $(0,3)$ such that
$f '(2) = b$ is a subspace of $\mathbb{R}^{(0,3)}$ if and only if $b = 0$. How to prove it?
Subspace of differentiable function
linear algebra
linear algebra
The set of differentiable real-valued functions $f$ on the interval $(0,3)$ such that
$f '(2) = b$ is a subspace of $\mathbb{R}^{(0,3)}$ if and only if $b = 0$. How to prove it?
Best Answer
First suppose that this set is a subspace of $\mathbb{R}^{(0,3)}$, then you know that it is closed under $+$ :
Now suppose that $b=0$. We need to prove that our set is non-empty and closed under $+$ and $.$ :