Polynomial Degree n Basis: 1,x,x^2...x^n

[Zhang Jiawen]

I knew that a polynomial of degree n has n+1 basis, i.e 1,x,x^2...x^n;
But what if a0=0,i.e the constant term is 0, like x^3+x, then what is the dimension and the basis? Is there only x(one dimension) as the basis?

 

[andrewkirk]

The basis is not a basis of a polynomial, it is the basis of a module of polynomials, which is a set with added algebraic structure.

Elements of modules and vector spaces do not have dimensions or bases. It is the module or vector space that has those things.

The representation of $x^3+x$ in the above basis is (0, 1, 0, 1, 0, ..., 0).

 

[HallsofIvy]

I knew that a polynomial of degree n has n+1 basis, i.e 1,x,x^2...x^n;
But what if a0=0,i.e the constant term is 0, like x^3+x, then what is the dimension and the basis? Is there only x(one dimension) as the basis?

Just drop "1" from the basis you give so the dimension is n.