2. Let C be the vector space of continuous functions and let V be the subspace spanned by the set E={cos(x), sin(x), cos(2x), sin(2x)}. Consider the linear transformation T : VV defined by T(f(x)) = f'(x), sending a function f(x) = V to its derivative. (a) Show that Ɛ is a basis for V. Hint: Plug in x = 0, π/2, π. (b) Find the matrix [T] of the transformation T with respect to the basis ε. (c) Show that [T] is a diagonal matrix. (d) Describe the linear transformation represented by the matrix [T]. That is, give L(f(x)) for the linear transformation LVV with [L] = [T]. [1234] -5 (e) Suppose f(x) = V has coordinates [f(x)] = with respect to the basis Ɛ. Compute the 100th π derivative of f. Your answer should be an element of V.

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Please solve for part B. Please show all work and steps.

A railway line connects Parnac station with Rougelle station. The distance between these two stations is 150 km. -a train departs from Parnac at 17:10 and stops at Bernouille station at 17:40. Bernern is 60 km from Parnac - He leaves Bernouille twenty minutes later and crosses the Sévillon train station at 18:30. The Sévillon train station is 45 km from Rougelle. He continued his journey at the same speed until Rougelle station. -Another train leaves Rougelle at 17:35 and arrives at Sévillon at 18h. After 10 minutes of waiting he leaves to reach Bernouille at 18:40 and stops 10 minutes. He finally joined Parnac at 19:20. "At what time and how far from Rougelle do the trains intersect?"