Answer to Question #52677 in Linear Algebra for ravish

Question

Let V be the set of all functions that are twice differentiable in R and S={cosx,sinx,xcosx,xsinx}. a)Check that S is a linearly independent set over R.(Hint:Considertheequation a0cosx+a1sinx+a2xcosx+a3xsinx. Put x=0,π,π/ 2 ,π /4 ,etc.and solve for ai.)
b)Let W=[S] and let T:V→V be the function defined by T(f(x))=d2/ dx2(f(x))+2d/ dx(f(x)). Check that T is a linear transformation on V.
c) check that T(W) is subset of W
d) write down the matrix of T on W w.r.t. the basis S
e) Is the matrix of the linear operator T non singular? Justify your answer