- #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!