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Thread: MATH2601 Higher Linear Algebra

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

    So that part makes sense now.

    How do we complete the proof if |H| = 1?

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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by leehuan View Post
    So that part makes sense now.

    How do we complete the proof if |H| = 1?
    That can only happen if a = e (the identity). Take a to be any other element in G (there must be at least one other element since G has prime order, which implies |G| is at least 2), and the result will follow (since |H| won't be able to be 1).
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    Re: MATH2601 Linear Algebra/Group Theory Questions



    The questions were



    For the first one I claimed it was true by using a uniqueness result


    and by pairing T(v) with 0. But I can't figure out why this argument does not work for the second one?

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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Why do you think that "uniqueness result" is true? Add anything orthogonal to x to y and you won't change the inner product of x with it.

    (Also the truth of b) is (perhaps surprisingly) dependent on the field your vector space is over.)
    Last edited by seanieg89; 6 Apr 2017 at 4:44 PM.
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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by leehuan View Post


    The questions were



    For the first one I claimed it was true by using a uniqueness result


    and by pairing T(v) with 0. But I can't figure out why this argument does not work for the second one?
    For a), we can do it like this: since T is from V -> V and < u, T(v) > = 0 for all u, v in V, for each v, just take u = T(v), and the result will follow.
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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Is there an intuitive explanation for this?

    Let T be a linear map on a finite-dimensional inner product space V

    Then T is an isometry iff T is unitary

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    Re: MATH2601 Linear Algebra/Group Theory Questions





    So the question is obviously easy first year stuff. I'd prove linear independence and then use dim(V) = B to deduce that it's a basis.

    However, for the linearly independence step



    I just want a validity check because I'm having second doubts. Mostly because no solutions made a remark on this.
    I differentiated w.r.t t and then subbed in t=2 to prove c1 = 0. (And then repeated this to show c2 = c3 = 0.) Is this ok?

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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by leehuan View Post




    So the question is obviously easy first year stuff. I'd prove linear independence and then use dim(V) = B to deduce that it's a basis.

    However, for the linearly independence step



    I just want a validity check because I'm having second doubts. Mostly because no solutions made a remark on this.
    I differentiated w.r.t t and then subbed in t=2 to prove c1 = 0. (And then repeated this to show c2 = c3 = 0.) Is this ok?
    Yeah, that's OK for showing linear independence.
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    Supreme Member seanieg89's Avatar
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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by leehuan View Post
    Is there an intuitive explanation for this?

    Let T be a linear map on a finite-dimensional inner product space V

    Then T is an isometry iff T is unitary
    In an inner product space, the norm is defined in terms of the inner product. This means that any operator that preserves the inner product will preserve the norm.

    Less obvious is the fact that in an inner product space, the inner product can be written in terms of the induced norm (*). Consequently anything that preserves the norm will preserve the inner product.

    Of course, you don't need to prove (*) in order to answer this particular question, but it kind of hits at the heart of the relationship between inner products and induced norms on a real/complex inner product space and is a good exercise.

    Note also that finite dimensionality is not required in any of these arguments.
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    Re: MATH2601 Linear Algebra/Group Theory Questions







    The question is an extension on c. How can I explain this answer from first-year calculus?

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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by leehuan View Post






    The question is an extension on c. How can I explain this answer from first-year calculus?




    Last edited by InteGrand; 23 Apr 2017 at 10:03 PM.
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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


    No.
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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by InteGrand View Post
    No.
    Okay good I agree. But what would be an easy method to generate a counterexample?

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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by leehuan View Post
    Okay good I agree. But what would be an easy method to generate a counterexample?
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    Re: MATH2601 Linear Algebra/Group Theory Questions

    I don't know much about Graph Theory and Group Theory but are they two different topics? Or Different names but same subjects?

    I can't be bothered Googling it
    | B Eng (Hons) | IB Mathematics SL | IB Mathematics HL | Australian Cricket | Casual University Statistics Tutor

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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by davidgoes4wce View Post
    I don't know much about Graph Theory and Group Theory but are they two different topics? Or Different names but Two dame subjects?

    I can't be bothered Googling it
    Two different topics.

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    Re: MATH2601 Linear Algebra/Group Theory Questions

    I was wondering if given the trace and the determinant of a matrix could you write down a unique matrix satisfying these conditions, or a simple formula for the family of matrices satisfying it?

    Mostly asking for the 2x2 case

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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by leehuan View Post
    I was wondering if given the trace and the determinant of a matrix could you write down a unique matrix satisfying these conditions, or a simple formula for the family of matrices satisfying it?

    Mostly asking for the 2x2 case
    No, the value for the trace and determinant of a real or complex matrix does not uniquely specify the matrix.



    (Note that for a 2x2 complex matrix, the trace and determinant will uniquely specify the eigenvalues of the matrix though.)
    Last edited by InteGrand; 8 May 2017 at 11:21 PM.
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    Re: MATH2601 Linear Algebra/Group Theory Questions

    The intuition of the question was this



    I then realised that B may share the same eigenvectors as A, and have eigenvalues equal to the square root of those of A. But I'm not sure where to proceed from there.

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    Re: MATH2601 Linear Algebra/Group Theory Questions

    Quote Originally Posted by leehuan View Post
    The intuition of the question was this



    I then realised that B may share the same eigenvectors as A, and have eigenvalues equal to the square root of those of A. But I'm not sure where to proceed from there.
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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


    No it need not be. Say V = R^2 and W1 = R^2 (= V) and W2 be the line (t, 0) (the x-axis). Take T to be a rotation map by 90 degrees counter-clockwise about the origin say. Then T is a linear map from V to V, so T(V) = T(W1) is a subspace of W1 = V = R^2, and W2 is a subspace of W1 (which is a subspace of V), but clearly W2 is not invariant under T (e.g. the point (1, 0) in W1 does not get mapped to a point in W2 by T; it gets mapped to (0, 1)).
    Last edited by InteGrand; 22 May 2017 at 11:20 PM.
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    Re: MATH2601 Higher Linear Algebra





    My approach thus far: Write



    Is this a dead end? Because I don't see how I can use what I know about T here
    Last edited by leehuan; 7 Jun 2017 at 9:13 PM.

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    Re: MATH2601 Higher Linear Algebra

    Quote Originally Posted by leehuan View Post




    My approach thus far: Write



    Is this a dead end? Because I don't see how I can use what I know about T here
    Claim: W := im(T) is such a subspace.

    Proof: Exercise.
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