Univariate Monomials form Infinite Basis for Univariate Polynomials on Complex Numbers

Last updated Nov 1, 2022

Let $P[\mathbb{C}]$ to be the polynomial vector space over $\mathbb{C}$. Then the Set of Monomials $S = {1, x, x^{2}, \dots}$ is an infinite Vector Space Basis for $P[\mathbb{C}]$.

Consider the Set $S = {1, x, x^{2}, \dots}$. We will show that $S$ is a Vector Space Basis for $P[\mathbb{C}]$. Observe that by definition, we can write any Polynomial as a Linear Combination of finitely many Monomials, and $S$ is the Set of all Monomials. Therefore $\text{span} S = P[\mathbb{C}]$, since The Subspace Span is the Set of all Linear Combinations.

On the other hand, suppose that there exists some nontrivial $c_{1}, \dots, c_{n} \in \mathbb{C}$ and $x^{a_{1}}, \dots x^{a_{n}} \in S$ for some $n \in \mathbb{N}$ so that $$c_{1} x^{a_{1}} + \cdots + c_{n} x^{a_{n}} = \mathbf{0}.$$ WLOG let $a_{1}, \dots, a_{n}$ be arranged in Increasing order (by Commutativity of $+$). Then, by definition of $\mathbf{0}$, $c_{1} x^{a_{1}} + \cdots + c_{n} x^{a_{n}}$ has infinitely many Polynomial Zeroes. However, the Fundamental Theorem of Algebra tells us that $c_{1} x^{a_{1}} + \cdots + c_{n} x^{a_{n}}$ has at most $a_{n}$ Polynomial Zeroes in $\mathbb{C}$ (since some $c_{i} \neq 0$ for $i \in [n]$) and $a_{n}$ is finite $\unicode{x21af}$. Therefore, we must have that $c_{1} = \cdots = c_{n} = 0$ and $S$ is a Linearly Independent Set.

Thus, $S$ is a Vector Space Basis. To see that it is an Infinite Set, consider $S’ = {x, x^{2}, \dots}$. Mapping $x^{n} \mapsto x^{n+1}$ for $n \in \mathbb{Z}_{\geq 0}$ gives us a Bijection between $S$ and $S’$ since:

1. For each $x^{k}$ for $k \in \mathbb{N}$, we have that $k-1 \in \mathbb{Z}_{\geq 0}$, so there is $x^{k-1} \in S$ that maps to $x^{k} \in S’$. So our mapping is a Surjection.
2. If $k, l \in \mathbb{N}$ so that $x^{k} \neq x^{l}$, then we have that $k \neq l$ and $x^{k-1} \neq x^{l-1}$. Therefore our mapping is an Injection.

Then, since $S’ \subsetneq S$ ($1 \not\in S’$), we have that $S$ is an Infinite Set by definition. $\blacksquare$

• Complex Numbers
• Subspace Span

1. Hoffman and Kunze - Linear Algebra - Sec 2.3 pg 43