The basis of the set of polynomials with degree at most n

Question

In my book it says that the basis of the set of polynomials of degree at most n ie $K[x]_{≤n}$ has the basis $1,x,x^2,...,x^n$. But arent $x,x^2,...$ a linear combination of each other? This confuses me since the set isnt a set of vectors here.

Answer 1

As said in a comment above, the polynomials $1,x,...x^n$ are linearly independent.

Maybe you have a confusion with the fact that $x^2 = x\times x$ (etc.). In this case, note that $x$ isn't a scalar so it can't be a coefficient of a linear relation.