Show that Pn(F) is generated by {1, x,... , x n }

Step 1 of 2

First to show that an arbitrary polynomial in \(P_{n}(F)\) is a linear combination of the polynomials in the set.

Consider \(p(x)=a_{0}+a_{1} x+a_{2} x^{2}+\ldots .+a_{n} x^{n} \in P_{n}(F)\) and \(\alpha_{0}, \alpha_{1}, \alpha_{2}, \ldots \ldots, \alpha_{n} \in F\).

Rewrite the equation.

\(\begin{aligned}
p(x) & =\alpha_{0} \cdot 1+\alpha_{1} \cdot x+\alpha_{2} \cdot x^{2}+\ldots .+\alpha_{n} \cdot x^{n} \\
a_{0}+a_{1} x+a_{2} x^{2}+\ldots .+a_{n} x^{n} & =\alpha_{0}+\alpha_{1} x+\alpha_{2} x^{2}+\ldots .+\alpha_{n} x^{n}
\end{aligned}\)