Consider the linear map T: C - C given by T(y) = y""+ 8y""+ 26y" + 40y' + 25y. An ODE expert assures you that every solution of T(y) = 0 can be written in the form Ce cos(t) + Ce sin(t) + Cate cos(t) + Cite sin(t) + Ce cos(t - ) + Ce sin(t - #) for some C , C,, C,, C , C C ER. Use this information to determine the dimension of the subspace H = kernel(T) c Co. In particular, . Give rigorous definitions of o kernel o dimension of a subspace Determine the dimension of H . Prove that your determination meets the conditions in your definition. Finally, explain whether it is possible to a) find two functions y , y, E C so that y,, y, are linearly dependent, but T(y ), T(y,) are linearly independent; b) find two functions y , yz E C so that y,, y2 are linearly independent, but T (y ), T (y,) are linearly dependent. For each, give an example, or explain why it is impossible. To save time and space in your explanations, you can assume/use: o The given map T: C - C is linear. o The ODE expert is correct. o The kernel of any linear map T: V - W is a subspace of the domain V. If you use other theorems or general facts (in particular equivalences between different definitions of dimension), state *and prove* them in detail. [You don't need more given facts for this project, but if you want to use more, then we want proof.]