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Thread: MATH2601 Higher Linear Algebra

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    Ancient Orator leehuan's Avatar
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    MATH2601 Higher Linear Algebra





    Hint was to use the Bezout property but I have no idea to use it. I assume it's related to proving the existence of an inverse because that was the only bit I had trouble proving. (Associativity is just a repeat proof and the identity element is obviously 1)
    ____________

    Side note - Am not sure why my threads had to be moved here.
    Last edited by dan964; 1 Mar 2017 at 4:43 PM.

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    Supreme Member seanieg89's Avatar
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    Re: Linear Algebra

    Suppose the integer a is a representative of a nonzero equivalence class in Z_p.

    Then a is coprime to p and hence am+pn=1 for some integers m and n.

    Projecting to equivalence classes we get [a][m]=1 (mod p). ([z] denotes the equivalence class in Z_p of the integer z.)

    I.e. every element of U_p has an inverse.
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    Supreme Member seanieg89's Avatar
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    Re: Linear Algebra

    In fact this argument straightforwardly generalises to tell us that the subset of Z_n consisting of only the equivalence classes coprime to n form a group w.r.t. multiplication.

    This yields Euler's theorem just as the version originally posted will yield Fermat's little theorem.
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    Supreme Member seanieg89's Avatar
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    Re: Linear Algebra

    Ps this isn't really linear algebra at all.
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    Taking a break! dan964's Avatar
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    Re: Linear Algebra

    Quote Originally Posted by seanieg89 View Post
    Ps this isn't really linear algebra at all.
    thread title updated to "Group Theory" so no longer misleading
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    Re: Group Theory

    I am well aware that group theory may not fall under the umbrella of 'linear algebra'. However, in my course (MATH2601), it is the first thing that they choose to teach, hence I set the thread up with that title. I would like my thread name back so I can reserve this thread for all of my math2601 questions. Thanks.

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    Re: Group Theory

    Quote Originally Posted by leehuan View Post
    I am well aware that group theory may not fall under the umbrella of 'linear algebra'. However, in my course (MATH2601), it is the first thing that they choose to teach, hence I set the thread up with that title. I would like my thread name back so I can reserve this thread for all of my math2601 questions. Thanks.
    When you double click on your thread when viewing from "whats new" you can change the name yourself.
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    Ancient Orator leehuan's Avatar
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    Re: Group Theory

    Quote Originally Posted by Rathin View Post
    When you double click on your thread when viewing from "whats new" you can change the name yourself.
    I know that. I just don't want to appear as though I am undoing a moderator action without permission.

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    Taking a break! dan964's Avatar
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    Re: Group Theory

    Quote Originally Posted by leehuan View Post
    I know that. I just don't want to appear as though I am undoing a moderator action without permission.
    its totally fine to edit the thread yourself
    I have renamed it again so that it is useful for other users of the site.
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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Just out of curiosity, what's an example of a vector space and/or a field which is defined in a way, that does not use the conventional means of addition and (scalar) multiplication?

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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Prove:

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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by Drsoccerball View Post
    Prove:
    Last edited by InteGrand; 7 Mar 2017 at 9:01 PM. Reason: Typo
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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by leehuan View Post
    Just out of curiosity, what's an example of a vector space and/or a field which is defined in a way, that does not use the conventional means of addition and (scalar) multiplication?
    there are probably some real valued matrix examples
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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Let G be a group with identity e. Prove that if x^2 = e for all x in G then G is abelian.

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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by Drsoccerball View Post
    Let G be a group with identity e. Prove that if x^2 = e for all x in G then G is abelian.




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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by InteGrand View Post




    Didn't notice that an element is it's own inverse thanks

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    Ancient Orator leehuan's Avatar
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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by Drsoccerball View Post
    Didn't notice that an element is it's own inverse thanks
    It's basically this



    Having used the associativity axiom and the definition of the identity element.

    Right-operating also works

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    Re: MATH2601 Linear Algebra/Group Theory Questions

    This is strictly for Leehuan's understanding, but here's a concrete example to flesh out for the non-standard addition operation question:



    and more generally:

    for a bijective function φ(x):

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    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Re: MATH2601 Linear Algebra/Group Theory Questions


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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by leehuan View Post
    Since V is finite dimensional, by definition V has a finite spanning set S, and this set S also spans W, so W is also finite dimensional.
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    Re: MATH2601 Linear Algebra/Group Theory Questions



    I have most of the proof covered up but I'm getting confused at the last bit.







    All I'm really stuck on is how to prove that if v \neq 0 why must alpha be equal to 0

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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by leehuan View Post


    I have most of the proof covered up but I'm getting confused at the last bit.







    All I'm really stuck on is how to prove that if v \neq 0 why must alpha be equal to 0
    leehuan likes this.

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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by leehuan View Post


    I have most of the proof covered up but I'm getting confused at the last bit.







    All I'm really stuck on is how to prove that if v \neq 0 why must alpha be equal to 0
    leehuan and kawaiipotato like this.

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    Re: MATH2601 Linear Algebra/Group Theory Questions







    So I commenced by stating from Lagrange's theorem that |H| is a factor of |G|. However since |G| is prime, the only possibilities are |H| = |G| or |H| = 1

    I'm looking at the |H| = |G| part. I want to deduce from |H| = |G| that H = G, so as H is clearly cyclic so must G. But how do I properly justify that H = G?

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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by leehuan View Post






    So I commenced by stating from Lagrange's theorem that |H| is a factor of |G|. However since |G| is prime, the only possibilities are |H| = |G| or |H| = 1

    I'm looking at the |H| = |G| part. I want to deduce from |H| = |G| that H = G, so as H is clearly cyclic so must G. But how do I properly justify that H = G?
    Well H has the same number of elements as G and is a subset of G, so H = G. (If S is a set that has only a finite number of elements, then the only subset of S with the same number of elements as S is S itself.)
    Last edited by InteGrand; 19 Mar 2017 at 11:02 PM. Reason: Typo
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