1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    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!

Vector spaces problem -linear algebra

  1. Jul 3, 2013 #1
    1. The problem statement, all variables and given/known data
    Hi guys , I have this problem ,well actually I don't understand the solution they provide , Here's the problem statement and the solution .
    May someone please explain the solution to me?? Thanks so much, Sorry for my bad english

    2. Relevant equations
    1.I understand that f'-af=0 and the kernel is the space of the solutions that satisfy that equation but I don't get what they do after that...why do they divide f(t) by e^(at)?
    2. why do they conclude that exists a constant c such that f(t)=ce^at??
  2. jcsd
  3. Jul 3, 2013 #2


    User Avatar
    Science Advisor

    The problem is this: we have the vector space of all infinitely differentiable functions, D is the differentiation operator.
    a) Find the kernel of the linear operator D- I.
    b) Find the kernel of the linear operator D- aI.

    It is simplest to solve (b) first, then take a= 1 to solve (a).
    If f is any function in the kernel of D- aI then, by definition of "kernel" we must have f'- af= 0.
    That is the same as df/dx= af which is a separable equation: [tex]df/f= adx[/tex]. Integrating both sides, ln(f)= ax+ d where c is the constant of integration. Taking the exponential of both sides [itex]f(x)= e^{ax+ c}= Ce^{ax}[/itex] where [itex]C= e^c[/itex].

    That is how I would have solved the problem. I suspect that your text, knowing that f must be an exponential, started from that:
    [tex]\left(\frac{f(x)}{e^{ax}}\right)'= \frac{f'(x)e^{ax}- f(x)ae^{ax}}{e^{2ax}}= \frac{(f'(x)- af(x))e^{ax}}{e^{2ax}}[/tex]
    by the product rule.

    And, because [itex]f'(x)- af(x)= 0[/itex], the right side is 0, the derivative of [itex]f(x)/e^{ax}[/itex] is 0 so that [itex]f(x)/e^{ax}[/itex] is a constant: [itex]f(x)/e^{ax}= C[/itex] so [itex]f(x)= Ce^{ax}[/itex].

    (And your English is excellent. Far better than my (put language of your choice here).
  4. Jul 3, 2013 #3
    Thank you so much HallsofIvy , I understood all your explanation . It's cool to know that there's still good people who like to help others .Greetings from Chile
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted