📂Linear Algebra

Polynomial Vector Spaces

Definition¹

Polynomial

A polynomial with coefficients from a field $F$ is defined for a non-negative integer $n$ as the following form.

$$ \begin{equation} f(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \cdots + a_{1}x + a_{0} \end{equation} $$

Here, each $a_{k} \in F$ is called the coefficient of $x^{k}$. If $a_{n}=a_{n-1}=\cdots=a_{0}=0$, then $f(x)$ is called a zero polynomial.

The degree of a polynomial is the largest power of $x$ with a coefficient $0$ not equal to zero in the form of $(1)$. For a non-zero constant $c$, the degree of $f(x) = c$ is $0$, and the degree of the zero polynomial is defined as $-1$ for convenience.

Polynomial Vector Space

Let $f, g$ be a polynomial with coefficients from $F$.

$$ f(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \cdots + a_{1}x + a_{0} $$

$$ g(x) = b_{m}x^{m} + b_{m-1}x^{m-1} + \cdots + b_{1}x + b_{0} $$

Without loss of generality, assume that $m \le n$ and define $b_{m+1}=b_{m+2}=\cdots=b_{n}=0$. Then $g(x)$ can be written as follows.

$$ g(x) = b_{n}x^{n} + b_{n-1}x^{n-1} + \cdots + b_{1}x + b_{0} $$

Now, let’s define the sum of two polynomials $f(x)$ and $g(x)$ as follows.

$$ f(x) + g(x) = (a_{n} + b_{n})x^{n} + (a_{n-1} + b_{n-1})x^{n-1} + \cdots (a_{1} + b_{1})x + (a_{0} + b_{0}) $$

For $c \in F$, define scalar multiplication as follows.

$$ cf(x) = ca_{n}x^{n} + ca_{n-1}x^{n-1} + \cdots + ca_{1}x + ca_{0} $$

Then, with respect to this addition and scalar multiplication, the set of all polynomials with coefficients from $F$ becomes a $F$-vector space, denoted by $P(F)$.

Description

The set of all polynomials with degree less than or equal to $n$ is denoted by $P_{n}(F)$. $P(F)$ includes all polynomials of any degree, whereas $P_{n}(F)$ includes all polynomials of degree less than or equal to $n$. Note that both are infinite sets and both are sets of polynomials (not sets of infinite series).

$$ P(F) = \left\{ a_{n}x^{n} + \cdots + a_{1}x + a_{0} : n \in \mathbb{N}, a_{i} \in F \right\} $$

$$ P_{n}(F) = \left\{ a_{k}x^{k} + \cdots + a_{1}x + a_{0} : 1 \le k \le n, a_{i} \in F \right\} $$

Moreover, while $P(F)$ includes polynomials of all degrees and is thus infinite-dimensional, $P_{n}(F)$ is of $n$ dimensions.