# Thread: Showing that P_3 is a subspace to P_n

1. Dear everyone,

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

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

3. Originally Posted by Cbarker1
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}

Originally Posted by Euge
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.

Originally Posted by Fernando Revilla
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.

6. Originally Posted by Cbarker1
What should be the first words that I should use in this verification? I am not good with proof-writing.
Do you know what "$\displaystyle 0\in P_3$" means?

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