1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Linear algebra- set of complex valued functions

  1. Apr 30, 2009 #1
    1. The problem statement, all variables and given/known data
    Let V be the set of all complex-valued functions f on the real line such that (for all t in R),
    f(-t)=f(t) with a bar on top (can't figure out Latex, sorry)
    The bar denotes complex conjugation.

    Give an example of a function in V which is not real-valued.


    2. Relevant equations



    3. The attempt at a solution
    Not quite sure what this means, just need a place to start really.
     
  2. jcsd
  3. Apr 30, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Write f(t)=u(t)+iv(t) where u and v are real valued functions. Equate the real and imaginary parts of both sides of your equation. What are the conditions on u(t) and v(t)?
     
  4. Apr 30, 2009 #3
    I didn't really know there were conditions on u(t) and v(t).
    To get an example, can I just make those fuctions whatever real valued function that I want, like t-2, and then plug it in?
     
  5. Apr 30, 2009 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    I mean the conditions you derive from f(-t)=conjugate(f(t)). If f(t)=u(t)+iv(t) isn't conjugate(f(t))=u(t)-iv(t)?
     
  6. Apr 30, 2009 #5
    Ohh, yeah, it does. So I have to make f(t) so that when the t's are (-t)'s, u(t)+iv(t) turns into u(t)-iv(t)?
     
  7. Apr 30, 2009 #6

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, so u(-t)+iv(-t)=u(t)-iv(t), right?
     
  8. Apr 30, 2009 #7
    Yep. Now I'm not sure where to go from here.
     
  9. Apr 30, 2009 #8

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Equate real and imaginary parts of both sides. Come on, help me out here.
     
  10. Apr 30, 2009 #9
    I'm sorry, I'm very confused here and I've never learned this before, so it's very frustrating.

    When you equate the real and imaginary parts, do you get
    u(-t)=u(t) and v(-t)=-v(t)?
     
  11. Apr 30, 2009 #10

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, exactly. Can you find two functions u and v so that u(-t)=u(t) and v(-t)=(-v(t)) and v is not equal to zero? So f is not real valued?
     
  12. Apr 30, 2009 #11
    So if u(t)=t^2+1 and v(t)=t^3-t,
    then f(t)=t^2 + 1 + i(t ^3-t)
    = it^3 + t^2 - it + 1
    Which is not real valued, so that's an example, correct?
     
  13. Apr 30, 2009 #12

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, u(t)=1 and v(t)=t works too. If you want to make it even simpler.
     
  14. Apr 30, 2009 #13
    Ah yes, that is much simpler.
    Thanks so much for the help, sorry I was very lost before. I appreciate your patience and guidance!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Linear algebra- set of complex valued functions
Loading...