Proving that a set is a set of generators

  • #1
274
1

Homework Statement


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.

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!
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,832
251
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:
 
  • #3
274
1
LOL. Thanks for the advice! I will remember it.

Thank you for looking over my work too!
 

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