Showing that P_3 is a subspace to P_n

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In summary, the conversation discusses the need to prove that ${P}_{3}$ is a subspace of ${P}_{n}$, which is the set of all polynomials of degree less than or equal to n. The conditions for ${P}_{3}$ to be a subspace are then provided as (1) 0\in P_3, (2) p,q\in P_3\Rightarrow p+q\in P_3, and (3) \lambda\in\mathbb{R},p\in P_3\Rightarrow\lambda p\in P_3. The conversation concludes with a question about how to start the proof and a clarification on what $P_3$ and $P
  • #1
cbarker1
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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 $n$?
 
  • #3
Cbarker1 said:
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
Euge said:
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
Fernando Revilla said:
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
Cbarker1 said:
What should be the first words that I should use in this verification? I am not good with proof-writing.
Do you know what "[tex]0\in P_3[/tex]" means?

(What you want to prove is not true for n= 0, 1, or 2!)
 
  • #7
I solved it.
 

FAQ: Showing that P_3 is a subspace to P_n

What is a subspace?

A subspace is a subset of a vector space that satisfies all of the properties of a vector space, including closure under addition and scalar multiplication. In other words, it is a smaller space within a larger space that still has all the same characteristics and operations as the larger space.

How do you show that P3 is a subspace of Pn?

To show that P3 is a subspace of Pn, you need to demonstrate that it satisfies the three properties of a subspace: closure under addition, closure under scalar multiplication, and contains the zero vector. In other words, you need to show that any polynomial of degree 3 or less added to another polynomial of degree 3 or less will result in a polynomial of degree 3 or less, that multiplying a polynomial of degree 3 or less by a scalar will result in a polynomial of degree 3 or less, and that the zero polynomial (with all coefficients equal to 0) is included in P3.

What is the difference between P3 and Pn?

P3 represents the vector space of all polynomials with a degree of 3 or less, while Pn represents the vector space of all polynomials with a degree of n or less. In other words, P3 is a specific subset of Pn.

Why is it important to show that P3 is a subspace of Pn?

Showing that P3 is a subspace of Pn is important because it helps us understand the structure and properties of both vector spaces. It also allows us to use theorems and operations from the larger vector space, Pn, on the smaller subset, P3.

Can P3 be a subspace of Pn if n is less than 3?

No, P3 cannot be a subspace of Pn if n is less than 3. This is because P3 represents polynomials of degree 3 or less, so if n is less than 3, it would not include all the polynomials in P3. In order for P3 to be a subspace of Pn, n must be greater than or equal to 3.

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