Let n be the vector space of all polynomials of degree n or

Let n be the vector space of all polynomials of degree n or less in the variable x. Let D2:42 be the linear transformation that takes a polynomial to its second derivative. That is, D2(p(x))=p(x) for any polynomial p(x) of degree 4 or less.

Let P_n| be the vector space of all polynomials of degree n| or less in the variable x|. Let D^2: P_4 rightarrow P_2|be the linear transformation that takes a polynomial to its second derivative. That is, D^2(p(x)) = p\"(x)| for any polynomial p(x)| of degree 4| or less. A basis for the kernel of D^2| is {|}|. Enter a polynomial or a comma separated list of polynomials. A basis for the image of D^2| is {|}|. Enter a polynomial or a comma separated list of polynomials.

Solution

Let P₄(x) = ax⁴+bx³+cx²+dx +e. Then P’(x) = 4ax³+3bx²+2cx +d and P”(x) = 12ax²+ 6bx +c.

The kernel of a linear transformation D² is the set of all polynomials P₄(x)   such that D²[P₄(x)] = 0

Then P”(x) = 12ax²+ 6bx +c = 0 so that a =b = c = 0. Then P₄(x) = dx +e . Thus a basis for Ker(D²) is { x,1}.

The image of D² is   { P₂ (x) P₂ : P₂ (x) = P”(x) for someP₄ (x) P₄ } = {12ax²+ 6bx +c P₂ : a,b,c are arbitrary scalrs/real numbers} . Thus, a basis for the image of D² is { x², x,1}.