## Polynomials as Vector Spaces

A vector space has not quite as much to do with vectors as you might think. A wide range of sets of things can be considered as vector spaces. Polynomials are usually considered as functions, but we cab treat them as vectors. This is because for a polynomial
$p(x)=a_0 +a_1 x+a_2 x^2 +a_3x^3+...+a_nx^n$
all the powers of
$x$
are linearly independent, so that for example
$x^2$
cannot be expressed in terms of higher or lower powers of
$x$
.
To express a polynomial of degree at most two as a vector,
write 1 as
$\begin{pmatrix}1\\0\\0\end{pmatrix}$

write
$x$
as
$\begin{pmatrix}0\\1\\0\end{pmatrix}$

write
$x^2$
as
$\begin{pmatrix}0\\0\\1\end{pmatrix}$

The polynomial
$2-3x+x^2$
can then be written as
$\begin{pmatrix}2\\-3\\1\end{pmatrix}$
.
All the usual rules of addition and scalar multiplication of vectors apply, so we can consider polynomials - of any degree - as a vector space.