I have several questions concerning the proof. I don't think I quite understand the details and motivation of the proof. Here is the proof given by our professor.
The space of polynomials $F[x]$ is not finite-dimensional.
Proof. Suppose $F[x] = \operatorname{span}\{f_1,f_2,\dots,f_n\}$
Let us choose a positive integer $N$ such that $N > deg (f_i)$ for all $i = 1,\dots,n$. As $\{f_1,f_2,\dots,f_n\}$ spans $F[x]$ we can find scalars $a_1, a_2, \dots,a_n$ such that $x^N = a_1f_1 + a_2f_2 + \cdots + a_nf_n$. Then the polynomial
$G(x) = x^N - a_1f_1 - a_2f_2 - \cdots - a_nf_n \equiv 0$
is a polynomial of degree $N$ which is identically zero. This is a contradiction since $G(x)$ cannot have more than $N$ roots.
Questions
- Why is $G(x)$ identically zero and what does it mean for it to be identically zero?
- After obtaining that $G(x) = x^N - a_1f_1 - a_2f_2 - \cdots - a_nf_n \equiv 0$, why must we have more than $N$ roots?
What is the motivation behind choosing $N$ such that $N > deg (f_i)$ for all $i = 1,\dots,n?$
What are the implications if we chose $N$ less than or equal to $f_i$ such that $f_i$ has the greatest degree? How will the proof fail in this case?
If there are any other important details and key insights that are worthy to be pointed out please let me know so that I can better my understanding of the argument presented.