Polynomials as Vector Spaces
\[p(x)=a_0 +a_1 x+a_2 x^2 +a_3x^3+...+a_nx^n\]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
To express a polynomial of degree at most two as a vector,
write 1 as
\[2-3x+x^2\]can then be written as
All the usual rules of addition and scalar multiplication of vectors apply, so we can consider polynomials - of any degree - as a vector space.