Final answer:
To find the basis for the kernel and polynomial transformations, we need to determine the polynomials that are mapped to the zero vector or polynomial. The polynomial basis can be found by choosing a set of polynomials that spans the space.
Step-by-step explanation:
To find the basis for the kernel of the linear transformation defined by a vector space of polynomials, we need to find the polynomials that are mapped to the zero vector. The kernel is the set of all polynomials that satisfy this condition.
To find the basis for the kernel of the linear transformation defined by a polynomial transformation, we need to find the polynomials that are mapped to the zero polynomial. The basis will be the set of all these polynomials.
The kernel transformation is the function that maps vectors to their corresponding kernel elements. To find the basis for the kernel transformation, we need to find the basis for the kernel and determine the transformation of each basis element.
To find the polynomial basis, we need to find a basis for the vector space of polynomials. This can be done by considering the different degrees of polynomials and choosing a set of polynomials that spans the space.