\[p(x)=a_0 +a_1 x+a_2 x^2 +a_3x^3+...+a_nx^n\]
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.