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I know that the standard basis for polynomials of degree n is

{${1,x,x^2,...,x^n}$}

but why is the 1 in there? After all, can't 1 be written as a linear combination, where

$c_1 = 1/x$

and

$1 = x*c_1 $

2 Answers 2

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$c_1=1/x$ is neither a scalar nor a polynomial.

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You have confused the notion of a basis, see here:

http://en.wikipedia.org/wiki/Basis_%28linear_algebra%29