Showing that P_3 is a subspace to P_n

[cbarker1]

Dear everyone,

I need to show that the $${P}_{3}$$ is a Subspace to $${p}_{n}$$. how to start the proof?

 

[Euge]

Could you clarify what $P_3$ and $P_n $ are? Also, what is the condition on $n$?

 

[Fernando Revilla]

I need to show that the ${P}_{3}$ is a Subspace to ${p}_{n}$. how to start the proof?

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 :
$\begin{aligned}&(1)\quad0\in P_3.\\
&(2)\quad p,q\in P_3\Rightarrow p+q\in P_3.\\
&(3)\quad \lambda\in\mathbb{R},p\in P_3\Rightarrow\lambda p\in P_3.
\end{aligned}$

 

[cbarker1]

Could you clarify what $P_3$ and $P_n $ are? Also, what is the condition on $n$?

$${P}_{n}$$ is the set of all polynomials of degree less than or equal to n. So $${P}_{3}$$ is the set of all polynomials of degree less than three.

 

[cbarker1]

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 :
$\begin{aligned}&(1)\quad0\in P_3.\\
&(2)\quad p,q\in P_3\Rightarrow p+q\in P_3.\\
&(3)\quad \lambda\in\mathbb{R},p\in P_3\Rightarrow\lambda p\in P_3.
\end{aligned}$

What should be the first words that I should use in this verification? I am not good with proof-writing.

 

[HallsofIvy]

What should be the first words that I should use in this verification? I am not good with proof-writing.

Do you know what "0\in P_3" means?

(What you want to prove is not true for n= 0, 1, or 2!)

 

[cbarker1]

I solved it.