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Math Help - Subspace of P_4.

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    Subspace of P_4.

    Pet P4 be the set of all polynomials of degree less than four.
    Is the set S of polynomials of even degree less than four a subspace of P4?

    I think that it is; however my text says it isn't.

    Here's why I think that it is.

    let v be an element of the Set described. Then v is an even degree polynomial of deg less than four. If we take a scaler and multiply v by it, we get another polynomial of the same deg before. So, v is in S. Satisfying closure one.

    Let v,x be in S, then consider v+x v+x is a polynomial with the highest even degree between them, and this will be less than 4, and it will be even. So, v+x is in S.

    What did I do wrong, if anything?
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    Quote Originally Posted by Chris11 View Post
    Pet P4 be the set of all polynomials of degree less than four.
    Is the set S of polynomials of even degree less than four a subspace of P4?

    I think that it is; however my text says it isn't.

    Here's why I think that it is.

    let v be an element of the Set described. Then v is an even degree polynomial of deg less than four. If we take a scaler and multiply v by it, we get another polynomial of the same deg before. So, v is in S. Satisfying closure one.

    Let v,x be in S, then consider v+x v+x is a polynomial with the highest even degree between them, and this will be less than 4, and it will be even. So, v+x is in S.

    What did I do wrong, if anything?
    What is the degree of the zero polynomial ?
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    zero, but 0=2x0, so zero is an even number
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    I guess that x^2-2x is in P4 and -x^2 is as well. But xthe sum of those 2 polynomials is -2x, and that's not even. So, I guess I sort of rushed it
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    Quote Originally Posted by Chris11 View Post
    zero, but 0=2x0, so zero is an even number
    Yes, but that describes all polynomials of the form a_0 + 0x + 0x^2 + 0x^3. The zero vector here is the zero polynomial which is defined to have degree of -\infty.

    Either way, I'm sure you can see that our set doesn't satisfy the conditions for it to be a subspace of \mathcal{P}_4
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    Quote Originally Posted by o_O View Post
    Yes, but that describes all polynomials of the form a_0 + 0x + 0x^2 + 0x^3. The zero vector here is the zero polynomial which is defined to have degree of -\infty.
    Can you give a reference for that? Every book I have ever seen says that the zero polynomial has degree 0.

    Either way, I'm sure you can see that our set doesn't satisfy the conditions for it to be a subspace of \mathcal{P}_4
    Well, actually, the reason for posting the question in the first place was that he couldn't!
    Last edited by HallsofIvy; February 20th 2010 at 10:59 AM.
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    Well, actually, I can. I didn't know that the degree of the zero polynomial was defined as -infinity. lol
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    Quote Originally Posted by HallsofIvy View Post
    Can you give a reference for that? Every book I have ever seen says that the zero polynomial has degree 0.
    I was taught that it was. The textbook I used was Axler's Linear Algebra Done Right. And wiki seems to agree too.

    But for a perhaps better reference, here's an entry about the zero polynomial on Wolfram: Zero Polynomial

    Looks like it's either undefined, -\infty, or -1. I'm pretty sure it can't be 0 though .. not that I'm an expert or anything.

    Well, actually, the reason for posting the question in the first place was that he couldn't!
    I meant that through the post about the degree of the zero polynomial and the OP's post which contained a counterexample, it should all be clear now.
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    you can also find that the zero polynomial has no degree.
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    Quote Originally Posted by o_O View Post
    I was taught that it was. The textbook I used was Axler's Linear Algebra Done Right. And wiki seems to agree too.

    But for a perhaps better reference, here's an entry about the zero polynomial on Wolfram: Zero Polynomial

    Looks like it's either undefined, -\infty, or -1. I'm pretty sure it can't be 0 though .. not that I'm an expert or anything.



    I meant that through the post about the degree of the zero polynomial and the OP's post which contained a counterexample, it should all be clear now.
    Thank you!
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