Dear everyone,
I need to show that the $${P}_{3}$$ is a Subspace to $${p}_{n}$$. how to start the proof?
Dear everyone,
I need to show that the $${P}_{3}$$ is a Subspace to $${p}_{n}$$. how to start the proof?
Could you clarify what $P_3$ and $P_n $ are? Also, what is the condition on $n$?
Perhaps you mean the vector space $P_n=\{p\in\mathbb{R}[x]:\deg p\le n\}.$ In such case prove the 3 conditions for $P_3$ to be a subspace :
$\begin{aligned}&(1)\quad0\in P_3.\\
&(2)\quad p,q\in P_3\Rightarrow p+q\in P_3.\\
&(3)\quad \lambda\in\mathbb{R},p\in P_3\Rightarrow\lambda p\in P_3.
\end{aligned}$