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Prove this is a Real Vector Space

  1. Feb 9, 2009 #1
    1. The problem statement, all variables and given/known data
    Let V be the real functions y=f(x) satisfying d^2(y)/(dx^2) + 9y=0.

    a. Prove that V is a 2-dimensional real vector space.
    b. In V define (y,z) = integral (from 0 to pi) yz dx. Find an orthonormal basis in V.

    3. The attempt at a solution
    part A:
    I integrated and got that f(x)= (-3/2)y^3 + Cy, C is a real number.
    It seems like I need to use dot product here. I don't know how, though.

    part B:
    Completely lost.
     
  2. jcsd
  3. Feb 9, 2009 #2

    Mark44

    Staff: Mentor

    First off, f(x) should be a function of x.
    Second, you made a mistake in solving your DE: the solutions to y'' + 9y = 0 are not what you show.

    What do you know about the characteristic equation of a linear DE?
     
  4. Feb 9, 2009 #3
    Oh, I never took a DE class. All I did was integrate it twice, and I didn't even realize it wasn't a function of x. What should I do?
     
  5. Feb 10, 2009 #4

    Mark44

    Staff: Mentor

    OK. You don't need to know how to solve the equation in order to verify that the set of all solutions to it is a vector space.
    You know that you have to verify a set of axioms involving addition and scalar multiplication, right?
     
  6. Feb 10, 2009 #5
    Yes..
    Under addition we must show it is well-defined and
    1. Associativity
    2. commutativity
    3. zero element
    4. inverses
    5. closed

    Under multiplication we need to show well-definedness and
    1. L and R distribution
    2. associativity
    3. id (unit)
    4. zero element
    5. closed

    do I have it covered here?
     
  7. Feb 10, 2009 #6

    Mark44

    Staff: Mentor

    Yes, those are the axioms that you have to verify.

    Let f, g, and h be elements of V. (What does that imply in regard to your DE?. IOW, what exactly does it mean for a function to be an element of V?)
    Let a and b be real numbers.

    To pick a couple, #4 in the addition group and #4 in the scalar multiplication group:

    If g is in V, is there another element h in V so that g + h = 0?
    If g and h are in V, is g + h in V?
     
  8. Feb 10, 2009 #7

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    The point was that you wrote f(x)= (-3/2)y^3 + Cy. "y" is not "x"! If you had had d^2y/dx^2= a function of x, then you could integrate twice but you cannot integrate an expression in "y" with respect to x when you don't know y as a function of x.

    Presumably you already know that addition of functions is associative and commutative, that f(x)= 0 for all x is the additive identity, etc and don't need to prove that here. All you really need to show is that if f(x) and g(x) are functions satisfying d^2f/dx^2+ 9f= 0 and d^2g/dx^2+ 9g= 0 then any linear combination of them, af(x)+ bg(x) for a and b any numbers, also satisfies that.
     
  9. Feb 10, 2009 #8
    Well how do I use the second derivative in helping me prove that V satisfies all the axioms?
     
  10. Feb 11, 2009 #9

    Mark44

    Staff: Mentor

    By using the theorems of differentiability. If f and g are differentiable functions, then so is f + g, and (f+g)' = f' + g'. Also, if a is a constant, then af is differentiable, and (af)' = af'. These ideas can be extended to the next higher derivative.
     
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