Space of Polynomials over a Field form a Vector Subspace of Function Vector Space

Last updated Nov 1, 2022

Let $F$ be a Field

We know from Functions to a Field form a Vector Space

Suppose $c \in F$ and $p, r \in P[F]$. Then there exist $n, m \in \mathbb{Z}{\geq 0}$so that$$\begin{align*} &p = c{0} + c_{1} x + \cdots + c_{n} x^{n}\\ &r = d_{0} + d_{1} x + \cdots + d_{m} x^{m}\\ \end{align*}$$ Without Loss of Generality we can take $n \geq m$ and then rewrite $$r = d_{0} + d_{1} x + \cdots + d_{m} x^{m} + d_{m+1} x^{m+1} + \cdots d_{n} x^{n},$$ taking $d_{m+1}, \dots, d_{n} = 0$. Then we get that $$\begin{align*} cp + r &= \sum\limits_{k=0}^{n}c c_{k}x^{k} + \sum\limits_{k=0}^{n} d_{k}x^{k}\\ &=\sum\limits_{k=0}^{n}(c c_{k} + d_{k})x^{k} \in P[F] \end{align*}$$

$\blacksquare$

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