Linear Algebra: Dimension of kernel

Question

Suppose that we have the vector space

$V=\{f/f:R\rightarrow R, \text{every class derivative is defined in R}\}$

and $φ: V\rightarrow W$ with $φ(f)=f+f'$ is linear. I want to find the dimension of the $Ker φ$ and if $Β\leq V$ with $Β\cap A=\{0\}$ to show that the restriction of $φ$ in $B$ is one to one. Any ideas please?

Answer 1

Let's $f \in V$ and $g \in \text{Ker}\{\varphi\} \subset V$, thus we have $\varphi(f+g) = \varphi(f)$

So we can write: $\varphi(f+g) = \varphi(f) + \varphi(g) = f +f' + g +g' = f +f' \Rightarrow g +g' = 0$

Which is an ODE with solution $g = ke^{-t}$. This spans the kernel which has dimension 1.

Answer 2

I agree with your solution. How can I prove now that the restriction of $\phi$ in $B$ is one to one?