Linear subspace of K[X]? I got the following task from my professor and I wanted to ask for advice from you.
Task: 


*

*$K$ is a field


I shall prove this statement
Prove that for every $v$ element of $K$ the set $I_v = \{f\,\text{element}\, K[X]/f(v) = 0\}$ is a linear subspace of $K[X]$!

I don't need to prove that much here just that: if p and q are polynoms, $p(v) = 0$ and $q(v) = 0$, $p + q$ is a polynom with $(p+q)(v) = 0$.
Same for the scalar multiplication.
Though I don't know how to write it down here.

If possible tell me if my thoughts are right and maybe complete my solutions if they are nearly right or give me useful hints?

We can do this using the One-Step Vector Subspace Test. Explicitly, let $f,g\in I_v$ and let $\lambda\in K$. Then
$$
(f+\lambda\cdot g)(v)=f(v)+\lambda\cdot g(v)=\bf 0+\lambda\cdot\bf 0=\bf 0+\bf0=\bf0
$$
Hence $f+\lambda\cdot g\in I_v$ and $I_v$ is a subspace of $K[X]$ as required.
The result also follows from an observation of @k.stm. Let $T:K[X]\to K$ be the linear map $T(f)=f(v)$. Then $I_v=\ker T$ and, of course, the kernel of any linear map is a subspace of its domain.
