Question

Let Coo denote the set of smooth functions, ie, functions f : R → R whose nth derivative exists, for all n. Recall that this is a vector space, where "vectors" of Coo are function:s like f(t) = sin(t) or f(t) = te, or polynomials like f(t)-t2-2, or constant functions like f(t) = 5, and more The set of smooth functions f (t) which satisfy the differential equation f"(t) +2f (t) -0 for all t, is the same as the kernel of the map T·CX → CX given by T(f)-f" +4f. (a) Show that T is a linear map by comparing T(af + bg) to aT(f) + bT(g) for any choice of scalars a, b in R and smooth functions f (t) and g(t) (b) Show that f (t) -sin(2t) is in the kernel of T. (You'll have to recall from calculus the chain rule and the derivatives of sine and cosine (c) Show that f(t) -cos(2t) is also in the kernel of T (d) Is the list of functions {sin(2t), cos(2t)} linearly independent or linearly dependent? Explain how you know (e) Suppose you are told that the kernel of T is two-dimensional. Using this information and your answer to (d) above, write the general form for the set of smooth functions f(t) which satisfy the differential equation f" 0. (f) Suppose you notice that If f(t) = t/4, then T(f) = f" + 4f = 0+4(t/4) = t. Use this information and your work above to describe the set of all smooth functions that satisfy the differential equation f"(t) + 4f(t) = t. Hint: Recall that, if T : V → W is a linear map between vector spaces with kernel ker(T) = Span(v1, ,呦, and you know T(w) = b, then the solution set for T(a) = b is x-w+sivi+..+ skVk : s, are scalars in R)

Answer 1

1 Answon iom J:kellt):うand more a (flt) .bg(り he smool..undro, a.b

0 defender]-( kp)ain how you know ay 1?neary 57n (o

2t I.t "จเดี่