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The set P of polynomials

  1. May 20, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider the set P4 of all real polynomials if degree <= 4.

    1)Prove that P4 is a subspace of the vector space of all real polynomials
    2)What is the dimension of the vector space P4. Prove answer by demonstrating a basis and verifying the proposed set is really a basis.


    2. Relevant equations
    3. The attempt at a solution

    1)Let the vector ##V = P={a_0+a_1x+a_2x^2+...+a_nx^n}## where the coefficients are real numbers

    let ##p(x)=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4## ##q(x)=b_0+b_1x+b_2x^2+b_3x^3+b_4x^4##

    Then [itex](p+q)(x) =p(x)+q(x)= (a_0+b_0)+(a_1+b_1)x+(a_2+b_2)x^2+(a_3+b_3)x^3+(a_4+b_4)x^4[/itex]

    [itex]kp(x)=(kp)(x)=ka_0+ka_1x+ka_2x^2+ka_3x^3+ka_4x^4[/itex]

    thus p+q and kp are in V....?

    2) Dimension is 5, ie we need 5 coefficients to have the basis linearly independent.
    How do I demonstrate a basis and verify it?

    Thanks
     
  2. jcsd
  3. May 20, 2012 #2

    Hurkyl

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    It sounds like you already have a guess how to find the coordinates of a polynomial with respect to a particular basis; you just haven't yet found the basis. What would the coordinates of a basis vector be?
     
  4. May 20, 2012 #3
    Well the definition of a basis is that the basis set has to be linearly independent and it spans the vector V
    So could we use ##(1,x,x^2,x^3,x^4)## since this is linearly independent.

    Would the coordinates be the coefficients of ##(a_0,a_1x,a_2x^2,a_3x^3,a_4x^4)##...?
     
  5. Aug 31, 2012 #4
    As you say the basis is a set. It is customary to use { } to denote a set. Take care not to make the assumption that you absolutely need a (... ,... , ) notation to denote vectors in this case c+bx+cx²+ ... is a vector.
     
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