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  1. MHB Apprentice

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    #1
    Dear everyone,

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

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

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    Fernando Revilla's Avatar
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    #3
    Quote Originally Posted by Cbarker1 View Post
    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 :
    $\begin{aligned}&(1)\quad0\in P_3.\\
    &(2)\quad p,q\in P_3\Rightarrow p+q\in P_3.\\
    &(3)\quad \lambda\in\mathbb{R},p\in P_3\Rightarrow\lambda p\in P_3.
    \end{aligned}$

  4. MHB Apprentice

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    #4 Thread Author
    Quote Originally Posted by Euge View Post
    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.

  5. MHB Apprentice

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    #5 Thread Author
    Quote Originally Posted by Fernando Revilla View Post
    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.

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    #6
    Quote Originally Posted by Cbarker1 View Post
    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!)

  7. MHB Apprentice

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    #7 Thread Author
    I solved it.

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