I know that a basis for the kernel is the map of the nullspace of the standard matrix for the linear transformation.

I found the nullspace for a particular problem to have a basis formed by the set

1/2

-3

-1/4

1

The original basis was the standard basis {1, x, x^2, x^3}.

Because I am working with polynomials, my professor said that the basis for a kernel must be polynomials. How do I change the nullspace basis given above into a polynomial?