Results 1 to 3 of 3

Math Help - Set of Polynomials

  1. #1
    Fel
    Fel is offline
    Newbie
    Joined
    Jun 2009
    Posts
    15

    Set of Polynomials

    Let V_n be the set of polynomials with real coefficients of degree at most n, that is a_n*x^n+...+a_0 V_n.

    a) Prove that {x^n, x^n-1,..., 1} is linearly independent.
    b) Show that span {x^n, x^n-1,..., 1} = V_n.
    c) Deduce that V_n is a vector space with basis {x^n, x^n-1,...,1}.
    d) Find the dimension of V_n.
    e) Show that the derivative is a linear map; d/dx: V_n --> V_n-1.
    f) Find the matrix associated to d/dx with respect to the bases for V_n and V_n-1 from part (c).

    ( _ denotes subscripts while ^ denotes superscripts)

    I'm not sure where to start with this one so if you guys can steer me in the right direction, it would be great. Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Nov 2008
    Posts
    34
    Hi!

    (a) How many zeros can a polynomial with degree n != 0 have?
    (b) definition of span
    (c) definition of a basis
    (d) definition of dimension
    (e) that follows from rules how to calculate the derivative
    (f) the (coordinates of the) images of the basis vectors form the columns of the associated matrix

    That is a good exercise to get used to the concept of a vector space. You just have to follow definitions so there should not be any problems.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    To show (a), an easy way is to suppose a polynomial is identically zero, and take its derivative until you get to a contradiction.

    Otherwise you can use the Vandermonde determinant but that's a little more complicated.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 7
    Last Post: April 7th 2011, 12:38 PM
  2. GCD of polynomials in Zn[x]
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: May 18th 2010, 06:22 AM
  3. Polynomials
    Posted in the Algebra Forum
    Replies: 3
    Last Post: May 16th 2010, 06:52 AM
  4. Replies: 7
    Last Post: January 8th 2010, 03:13 AM
  5. Replies: 5
    Last Post: November 29th 2005, 03:22 PM

Search Tags


/mathhelpforum @mathhelpforum