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Thread: Linear Algebra Marathon & Questions

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    Supreme Member seanieg89's Avatar
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    Linear Algebra Marathon & Questions

    Linear Algebra Marathon & Questions
    This is a marathon thread for linear algebra. Please aim to pitch your questions for first-year/second-year university level maths. Excelling & gifted/talented secondary school students are also invited to contribute.

    (mod edit 7/6/17 by dan964)

    ===============================
    To accompany the corresponding calculus thread.

    First question (spectral theorem, familiarity with dot product recommended):




    Last edited by dan964; 7 Jun 2017 at 4:41 PM.

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    Rambling Spirit
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    Re: First Year Linear Algebra Marathon

    Last edited by InteGrand; 30 Oct 2016 at 5:34 AM.

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    Ancient Orator leehuan's Avatar
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    Re: First Year Linear Algebra Marathon

    Quote Originally Posted by InteGrand View Post

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    Taking a break! dan964's Avatar
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    Re: First Year Linear Algebra Marathon

    A nice and simple question for first year uni students:
    Prove the Cauchy-Schwarz inequality.

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    -insert title here- Paradoxica's Avatar
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    Re: First Year Linear Algebra Marathon

    Quote Originally Posted by dan964 View Post
    A nice and simple question for first year uni students:
    Prove the Cauchy-Schwarz inequality.
    This is not a first year level problem...

    unless you're asking for the vector form of the inequality.

    For the sum form of the inequality, the proof is trivial.

    Consider the following quadratic equation in x:



    In future, you should not omit detail, unless the context makes it clear. Which for this one, probably means vector form.
    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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    Taking a break! dan964's Avatar
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    Re: First Year Linear Algebra Marathon

    Quote Originally Posted by Paradoxica View Post
    This is not a first year level problem...

    unless you're asking for the vector form of the inequality.

    For the sum form of the inequality, the proof is trivial.

    Consider the following quadratic equation in x:



    In future, you should not omit detail, unless the context makes it clear. Which for this one, probably means vector form.
    Yes I was asking for inner product spaces, that is for vectors.

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    Ancient Orator leehuan's Avatar
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    Re: First Year Linear Algebra Marathon








    Quote Originally Posted by Paradoxica View Post
    unless you're asking for the vector form of the inequality.
    I did not even know that there was a sum form until doing past papers for 1251. Then I had to figure out why the sum and vector forms were equivalent.
    Last edited by leehuan; 22 Nov 2016 at 5:50 PM.

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    -insert title here- Paradoxica's Avatar
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    Re: First Year Linear Algebra Marathon

    Quote Originally Posted by leehuan View Post









    I did not even know that there was a sum form until doing past papers for 1251. Then I had to figure out why the sum and vector forms were equivalent.
    olympiad kids be like that's the second thing I would have thought of...

    first is power mean inequality, no less.
    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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    Ancient Orator leehuan's Avatar
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    Re: First Year Linear Algebra Marathon

    Quote Originally Posted by Paradoxica View Post
    olympiad kids be like that's the second thing I would have thought of...

    first is power mean inequality, no less.
    Pretty sure this question is too elementary for olympiad level

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    Supreme Member seanieg89's Avatar
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    Re: First Year Linear Algebra Marathon

    Quote Originally Posted by leehuan View Post









    I did not even know that there was a sum form until doing past papers for 1251. Then I had to figure out why the sum and vector forms were equivalent.
    Pretty sure you mean min. Also note that it looks mildly messier for complex inner product spaces but the same argument still works. (replace your 2a.b with ).

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    Supreme Member seanieg89's Avatar
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    Re: First Year Linear Algebra Marathon


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    Re: First Year Linear Algebra Marathon


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    Supreme Member seanieg89's Avatar
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    Re: First Year Linear Algebra Marathon



    (This result gives the first glimpse of why analysis of infinite dimensional vector spaces is far more subtle than that on finite dimensional vector spaces. A sequence can converge with respect to one norm but not with respect to another.)
    Last edited by seanieg89; 16 Nov 2016 at 3:17 PM.

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    Re: First Year Linear Algebra Marathon

    If you managed to prove the Cayley-Hamilton theorem for complex matrices using a property of complex matrices (like that diagonalisable complex matrices form a dense set), would this automatically also imply the theorem for any field (or commutative ring), because the theorem is basically a bunch of algebraic identities that hold for any given matrix, and multiplication and addition etc. will all behave the same way regardless of what field the entries of the matrix come from?

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    Supreme Member seanieg89's Avatar
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    Re: First Year Linear Algebra Marathon

    Quote Originally Posted by InteGrand View Post
    If you managed to prove the Cayley-Hamilton theorem for complex matrices using a property of complex matrices (like that diagonalisable complex matrices form a dense set), would this automatically also imply the theorem for any field (or commutative ring), because the theorem is basically a bunch of algebraic identities that hold for any given matrix, and multiplication and addition etc. will all behave the same way regardless of what field the entries of the matrix come from?
    Would probably depend on exactly how you proved it. Most proofs would be valid for arbitrary fields, but if you did it using some very special facts about C, I don't know if it is any easier to pass from Cayley Hamilton for C to Cayley Hamilton for F than it is to just prove Cayley Hamilton for F from scratch.

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    Re: First Year Linear Algebra Marathon

    Quote Originally Posted by seanieg89 View Post
    Would probably depend on exactly how you proved it. Most proofs would be valid for arbitrary fields, but if you did it using some very special facts about C, I don't know if it is any easier to pass from Cayley Hamilton for C to Cayley Hamilton for F than it is to just prove Cayley Hamilton for F from scratch.

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    Loquacious One Drsoccerball's Avatar
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    Re: First Year Linear Algebra Marathon

    Ill take a good look at this thread along with the advanced marthons while im overseas goodluck people

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    Supreme Member seanieg89's Avatar
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    Re: First Year Linear Algebra Marathon

    Quote Originally Posted by InteGrand View Post
    I understood what you were asking, but the arbitrary field proofs of CH that I know of are rather short, so it seemed an odd detour to go via the complex special case.

    In any case, I think you can make your idea rigorous but it takes a bit more care and precision than in your paragraph. I think something like my argument below would work, but forgive any sloppiness as my algebra is quite rusty.

    Eg we would first observe that for an arbitrary matrix A over an arbitrary field K, the entries of p_A(A) are all fixed integer polynomials in the entries of A. We can map these integer polynomials to polynomials with coefficients in K by mapping the coefficients to K via the homomorphism m |-> 1+1+...+1 (m ones). The entries of p_A(A) are then given by these image polynomials evaluated at the entries of A, which are in K.

    Note however that this polynomial homomorphism is not injective, because 2x and 0 both get regarded as the zero polynomial in (Z/2Z)[X] for instance.

    However, in the complex case (in fact in arbitrary characteristic zero case), this mapping is injective. Hence, if the matrix has entries in C, then P_A(A) has entries that are integer polynomials in the entries of A, and if these polynomials vanish for all choices of entries, this implies they have all coefficients equal to zero. By injectivity, this implies that the the original integer polynomials have all coefficients zero and hence the entries of p_A(A) will be zero for arbitrary K.

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    Re: First Year Linear Algebra Marathon


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    -insert title here- Paradoxica's Avatar
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    Re: First Year Linear Algebra Marathon

    Prove the following, for any positive integer n:



    good luck. you'll need it.
    Last edited by Paradoxica; 22 Nov 2016 at 11:03 PM.
    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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    Kosovo is Serbian RenegadeMx's Avatar
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    Re: First Year Linear Algebra Marathon

    i swear some of the questions here arent even first year
    Drsoccerball and leehuan like this.



    Quote Originally Posted by Shadowdude View Post
    other bos-ers have a penis pic sharing club and i'm debating whether i want in on it or not


    ~just so i belong~
    Quote Originally Posted by Shadowdude View Post
    who cares about your timetable - you should be going into uni every day to study and do whatever regardless

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    -insert title here- Paradoxica's Avatar
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    Re: First Year Linear Algebra Marathon

    Quote Originally Posted by RenegadeMx View Post
    i swear some of the questions here arent even first year
    amazing.
    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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    Loquacious One Drsoccerball's Avatar
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    Re: First Year Linear Algebra Marathon

    Quote Originally Posted by RenegadeMx View Post
    i swear none of the questions here are first year
    ftfy.

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    Supreme Member seanieg89's Avatar
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    Re: First Year Linear Algebra Marathon

    First year courses have different content at different universities, and lecturers who write exams for these courses are way less constrained by things like syllabi than the people who write HSC exams.

    Think of this thread as a place to post questions accessible by a first year student (on the knowledge front), with any terminology that would not be standard in all first year courses clarified.

    In this sense it is similar to the advanced mx2 marathon, just you are a little less handcuffed re: the methods you can use and a higher standard of rigour is expected. And as always for these threads, I think the ideal question is more demanding on the ingenuity/creativity/intuition front than on the knowledge front.
    Paradoxica likes this.

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    -insert title here- Paradoxica's Avatar
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    Re: First Year Linear Algebra Marathon

    Quote Originally Posted by seanieg89 View Post
    First year courses have different content at different universities, and lecturers who write exams for these courses are way less constrained by things like syllabi than the people who write HSC exams.

    Think of this thread as a place to post questions accessible by a first year student (on the knowledge front), with any terminology that would not be standard in all first year courses clarified.

    In this sense it is similar to the advanced mx2 marathon, just you are a little less handcuffed re: the methods you can use and a higher standard of rigour is expected. And as always for these threads, I think the ideal question is more demanding on the ingenuity/creativity/intuition front than on the knowledge front.
    Amen
    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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