Two vector spaces being isomorphic

  1. sadly not been able to put much effort into this one! was a lecture i missed towards the end of term and didnt get the notes on it, however here is the question.
    for K>or equal to 1 let Pk denote the the vector space of all real polynomials of degree at most k. For which value of n is Pk isomorphic to Rn. Give a brief reason for your answer.

    Now from what i have found on two vector spaces being isomorphic they need to have equal dimensions (dimu=dimv) so knowing that we have dimRn)=n is as far as i have got. Not really understanding this one, surely they would be isomorphic at any value of n as long as it's between 1 and k????
  2. jcsd
  3. Re: isomorphic

    How many basis vectors do you need to span P_k?
  4. HallsofIvy

    HallsofIvy 41,260
    Staff Emeritus
    Science Advisor

    Re: isomorphic

    For example, P1 is the space of all first degree polynomials which can be written in the form ax+ b which, in turn, can be mapped to (a,b) in R2.
    Last edited: Jan 5, 2009
  5. Re: isomorphic

    so for a second degree polynomial you would have ax2 +bx +c so is the answer just n-1 because the dimension of a polynomial is always one higher than the degree?
  6. Dick

    Dick 25,913
    Science Advisor
    Homework Helper

    Re: isomorphic

    A single polynomial doesn't have a 'dimension'. The point is that the set of ALL degree two or less polynomials can be represented as linear combinations of the three linearly independent functions 1, x and x^2. They form a basis. The 'dimension' of a space is the number of elements in a basis.
  7. Re: isomorphic

    Ok, so the basis formed by the polynomial is always going to be one higher than the degree of the polynomial, therefore value of n that will make Pk isomorphic to n has to be k+1? (crosses fingers and preys)
  8. Dick

    Dick 25,913
    Science Advisor
    Homework Helper

    Re: isomorphic

    Preys? I think you want to pray. Sure, P_k has dimension k+1. R^n has dimension n. Two finite dimensional vector spaces are isomorphic if they have the same dimension.
  9. Re: isomorphic

    thanks, thats much simpler than it first looked.
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