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!
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!
via Physics Forums RSS Feed http://www.physicsforums.com/showthread.php?t=708152&goto=newpost
0 commentaires:
Enregistrer un commentaire