How about you start by showing that D is a linear transformation?
Really stuck on this problem please help.
Let V be the vector space over R spanned by the functions,
e0(x)=1, e1=sin(x), e2(x)=cos(x), e3=sin(2x), e4(x)=cos(2x)
Furthermore consider the map D:V->V, f->df/dx,
where df/dx denotes the derivatives of f.
(a)Show that D is a linear transformation and determine the kernel ker(D) as well as the image imD.
(b)Show that the set {e0,e1,e2,e3,e4} is a basis for V.
(c)Determine a matrix representation of the linear transformation D.