Thread: Linear Transformation and Matrices #2.

Let V be the vector space of real-valued continuous functions with ordered bases S={sin t, cos t} and consider T={sin t - cos t, sin t + cos t}, another ordered basis for V. Find the representation of the linear operator L: V --> V defined by L(f) = f' with respect to:
(a) S
(b) T
(c) S and T
(d) T and S

again, what you want to do is examine what L does to the elements of S, or T.

the whole point of a basis, is that every element in V is a linear combination of basis elements. and L is a linear map, so determining L's action on basis elements determines L completely.