Simply asking how can someone show that the vector space $V$ of all polynomials on a field, say $K$ cannot be generated with any finite set of vectors?
I don't know where to tackle the problem. :(
Thank you.
Simply asking how can someone show that the vector space $V$ of all polynomials on a field, say $K$ cannot be generated with any finite set of vectors?
I don't know where to tackle the problem. :(
Thank you.
Suppose that it could be, for vectors (polynomials) $p_1,\ldots, p_n$. Let $m=\text{max}\{\text{deg}(p):p=a_1p_1+\ldots+a_np_n:a_1,\ldots,a_n\in K\}.$ Surely a polynomial of degree $m+1$ exists in $V$. This is a contradiction.
Another possibility is to show that the infinite family $\{X^n : n\geq 0\}$ is linearly independent.