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Homework Help: Linear algebra basis/dimensions

  1. Jan 20, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the dimensions and basis of the following vector space V over the given feild K:
    a) V is the set of all polynomials over R (real) of degree at most n and whose coefficients add to 0, K = R (real numbers)
    b) K = R (real), and V is the set of functions from R to R which are solutions of the differential equation: d2f/dx2 - 9f = 0

    2. Relevant equations
    definition of basis, spanning and dimensions

    3. The attempt at a solution
    for part a) i think the dimetions is N and the basis is all the factors

    for part b) i think it has 2 dimentions and the basis is df/dx - 3 and df/dx +3
  2. jcsd
  3. Jan 20, 2009 #2


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    Sort of. But what do you mean by 'factors' and df/dx-3 isn't a function from R->R. It's a differential equation. Can you spell out exactly what a basis is in each case?
  4. Jan 21, 2009 #3


    Staff: Mentor

    I might be wrong, but it appears that you "factored" this differential equation:
    to get this:
    One problem with that is that the first DE had -9f, not -9.
  5. Jan 22, 2009 #4
    By basic i mean something which is linear independent and spans V

    and by factors i mean suppose a1, a2, a3 ... an were the roots, then the factors would be (x - a1), (x - a2), (x - a3) ... (x - an) I suppose i should specfy the Real roots
  6. Jan 22, 2009 #5


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    Exactly. In the first case you should specify N polynomials and in the second case you should specify two functions.
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