## 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.