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The set of all polynomials of degree greater than three together with the zero polynomial in the vector space $P$ of all polynomials with coefficients in $\Bbb{R}$.

I thought I understood generally how to do this but my book (Linear Algebra: Fraleigh, Beauregard, Wesley 1995) explains how to determine whether the subset is a subspace of the vector space. It seems like it already assumes the set is a subset.

  1. How do I determine if a set is a subset of a vector space? Is it with the 8 axioms of vector addition and scalar multiplication?

  2. How do I read this problem specifically? I'm not really sure how to write these out in set notation.

This isn't homework perse. I am studying for my final tomorrow though.

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    $\begingroup$ Well, it's obviously a subset, since every polynomial of degree $>3$ is a polynomial. Did you mean to ask if it is a $\textit {subspace} $? If so, what about sums like $x^3+(-x^3+1)$? $\endgroup$
    – lulu
    Commented May 3, 2020 at 19:00
  • $\begingroup$ it is not a subspace since it is not closed under addition, for example: $\left(x^{3}+x^{2}\right)-x^{3}=x^{2}$ $\endgroup$
    – BinyaminR
    Commented May 3, 2020 at 19:03
  • $\begingroup$ Yeah sorry I meant subspace $\endgroup$ Commented May 3, 2020 at 20:25

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To check that a subset is a subspace you need to check three axioms:

1.Closed under addition

2.Closed under scalar multiplication

3.Non empty

In your case is very simple since $deg((x^3+1)+(-x^3))=deg(1) <3$

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  • $\begingroup$ Yeah I get the subspace axioms. I was just having a hard time reading the problem itself. Also, how do you come up with something like that without spending hours and hours just guessing different polynomials. $\endgroup$ Commented May 3, 2020 at 20:34
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$P = \mathbb{R}[x]=\{a_0+a_1x+\ldots+a_nx^n \mid n\in \mathbb{N}, a_i\in \mathbb{R}\}.$

$S = \{\mathbf{0}\} \cup\{p\in P \mid \deg(p) > 3 \}.$

$S$ is a subset of $P$ $\iff \forall s\in S. \ s \in P.$

Take $s \in S.$ Then $s=\mathbf{0}$ or $s\in P$ with $\deg(p) > 3.$

So you just have to show that $\mathbf{0} \in P.$

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  • $\begingroup$ is $\Bbb{R}|x|$ the set of all polynomials with a real coefficient? If so, can you explain how? I thought it would have been something like: $\{a_01, a_1x, a_2x^2, ... \}$ for $a_i \in \Bbb{R}$ $\endgroup$ Commented May 3, 2020 at 20:30
  • $\begingroup$ I expanded the definition of $\mathbb{R}[x]$. Notice that this is an answer to your first question. $\endgroup$
    – Riccardo
    Commented May 3, 2020 at 20:35
  • $\begingroup$ Thanks. So I was kind of right in my assumption. I'm guessing that using x is just simpler since x could be set equal to any mono / polynomial expression? $\endgroup$ Commented May 3, 2020 at 20:38
  • $\begingroup$ One remark: $\mathbb{R}[x]$ is not the set of polynomials with $a$ real coefficient, but the set of polynomials with real coefficients, i.e of the form I wrote in the answer. $x$ is just a variable and you can set it to any real value, but not to mono/polynomial expressions. $\endgroup$
    – Riccardo
    Commented May 3, 2020 at 20:43
  • $\begingroup$ I gotcha. I didn't think it was something like $a(1+x+x^2+...+x^n)$ I only said monomial because out side of this problem it could be set to that. But since this problem specified polynomials, then monomials would not be included. Sorry for confusing my understanding. $\endgroup$ Commented May 3, 2020 at 20:55

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