Answer to Question #52686 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: Consider the equation a0cosx+a1sinx+a2xcosx+a3xsinx. Putx=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)⊂W.(7)d) Write down the matrix of T on W w.r.t. the basis S.(2)e)Is the matrix of the linear operator T non-singular? Justify your answer.