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Proving that a set is a set of generators

  1. Sep 1, 2013 #1
    1. The problem statement, all variables and given/known data
    I want to show that the set
    $$
    <1,x,x^2,\cdots ,x^n>
    $$
    forms a basis of the space
    $$
    P_{n}
    $$
    where
    $$ P_{n} $$ contains all polynomial functions up to fixed degree n.

    3. The attempt at a solution
    I have already shown that the set
    $$
    <1,x,x^2,\cdots ,x^n>
    $$
    is linearly independent and now I want to show that this set is a set of generators for $$ P_{n}.$$

    Take any
    $$
    f\in P_{n}.
    $$ Let
    $$
    <\alpha_{0},...,\alpha_{n}>$$
    represent the coefficients of $$ f.$$ Then since
    $$
    \alpha_{0}\cdot 1=\alpha_{0},...,\alpha_{n}\cdot x^{n}=\alpha_{n}x^{n}
    $$
    adding these up gives us
    $$
    \alpha_{0}+\cdots+\alpha_{n}x^{n}=f.
    $$

    Is that correct or am I missing something? Thanks!
     
  2. jcsd
  3. Sep 1, 2013 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi DeadOriginal! :smile:

    (use # instead of $ and it won't start a new line every time! :wink:)

    Yes, that looks fine, except I think you can shorten it a little:

    you can say that by definition, any f in Pn is of the form ##
    \alpha_{0}+\cdots+\alpha_{n}x^{n}## :wink:
     
  4. Sep 1, 2013 #3
    LOL. Thanks for the advice! I will remember it.

    Thank you for looking over my work too!
     
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