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Vector space questions

  1. Oct 22, 2014 #1
    I'm having trouble with a couple of things written in some notes I'm reading.

    Firstly, in stating examples of vector spaces, they say
    Trigonometric polynomials - Given n distinct (mod 2π) complex constants λ1,...,λn, the set of all linear combinations of enz forms an n-dimensional complex vector space.

    Now does this even make sense? I feel as though it is trying to say that we can have enz for integer λn as the basis for the vector space of functions, i.e like Fourier series, but I'm not sure really.

    Secondly, for the complex vector space defined by the vectors in the 2D plane
    |r,φ>=(rcosφ,rsinφ)
    where vector addition is as usual and scalar multiplication follows the rule
    α|r,φ>=||α|r,φ+argα>
    I'm having trouble proving that
    α(|a>+|b>)=α|a>+α|b>
    and
    (α+β)|a>=α|a>+β|a>

    For the second one, I do
    (α+β)|a>=(α+β)|r,φ>
    =||(α+β)|r,φ+arg(α+β)>
    and I have no idea what to do next, as expanding the modulus/argument doesn't seem possible. Not sure about the second either.
     
  2. jcsd
  3. Oct 22, 2014 #2

    Mark44

    Staff: Mentor

    The λi's are complex, not integer.
    Show us where you've gotten on these. Start with the left side and expand it using the rule above for scalar multiplication.
     
  4. Oct 22, 2014 #3

    RUber

    User Avatar
    Homework Helper

    For thinking of them as a basis set, first consider what it would take to normalize them.
    Then, since the lambdas are distinct, mod 2pi, it can be shown that they are also orthogonal.
    This is similar in concept to the fourier basis, but with complex lambdas.
    For the last part, you should be able to put it back into complex exponential form and demonstrate the properties you are looking for.
     
  5. Oct 22, 2014 #4
    Yeah, I know that's what it says, but I just don't understand how that is the case... A quick google of trigonmetric polynomials shows them to be the functions
    sin(mx), cos(mx) or exp(imx) for integer m...

    I've started off one of them at the bottom, but can't progress...
     
  6. Oct 22, 2014 #5
    Hmm I think I'm having trouble understanding what it means by mod(2pi), especially with the lambdas being complex. Could you explain please?
     
  7. Oct 22, 2014 #6

    Mark44

    Staff: Mentor

    ##\pi/6## and ##13\pi/6## are in the same congruence class, modulo ##2\pi##
     
  8. Oct 22, 2014 #7
    Ok, so what happens if we have (13π/6)+2i, as we are allowed complex numbers? Is this (π/6)+2i?
     
  9. Oct 22, 2014 #8

    Mark44

    Staff: Mentor

    It might be easier to start with the right side, α|a>+α|b>.
    You could write the vectors a and b as (r1cosφ1) and (r2cosφ2). According to the scalar multiplication rule above, what is αa? What is αb?
     
  10. Oct 22, 2014 #9

    Mark44

    Staff: Mentor

    I believe so.
     
  11. Oct 22, 2014 #10

    RUber

    User Avatar
    Homework Helper

    Notice that the imaginary part of ##\lambda## will turn into your real part in the exponential, so the mod 2pi only needs to apply to the real part of ##\lambda## to prevent duplication in the imaginary part, since sin(x)=sin(x+2pi).
     
  12. Oct 22, 2014 #11
    Ok, so are these actually trigonometric polynomials - wikipedia seems to claim they are just sin(mx), cos(mx), exp(imx) for integer m.

    I'm confused about the whole mod(2pi) thing - why don't we just say we have all these lambdas and they must be between zero and 2pi?

    So the vector space produced by linear combinations of these will be any function?
     
  13. Oct 22, 2014 #12
    αa=[(|α|r1)cos(φ1+argα),(|α|r1)sin(φ1+argα)],
    αb=[(|α|r2)cos(φ2+argα),(|α|r2)sin(φ2+argα)]
    I did try going down this route and things got very messy...

    Anyway,
    αa+αb=[(|α|r1)cos(φ1+argα)+(|α|r2)cos(φ2+argα),(|α|r1)sin(φ1+argα)+(|α|r2)sin(φ2+argα)]
     
  14. Oct 22, 2014 #13

    Mark44

    Staff: Mentor

    I'm having some trouble with your notation. Apparently, you're using this notation for a vector: |r, φ>.
    Here, you're writing vectors as |a>. Where is the angle?
     
  15. Oct 22, 2014 #14
    Well |a> is just the vector, but it has an associated modulus and argument so I can write |a>=|r,φ>.
     
  16. Oct 22, 2014 #15

    Mark44

    Staff: Mentor

    This stuff would be easier to read if you put in some spaces. The expressions you have written don't have a single space anywhere.

    The natural thing to do would be to use the sum formulas for sine and cosine, and see if you can get some simplification from doing that.
     
  17. Oct 22, 2014 #16

    RUber

    User Avatar
    Homework Helper

    Since pi is irrational the the set of integers times x is equivalent to the set of distinct points in the range 0 to 2pi.
    In any case, whether you use mx or lambda x, you end up with the same vector space.
     
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