2. Let there be the set V which is defined to be V = y ( t ) | d - y (t ) + y(t) = 0, t E [0, 27], y(0) = yo, = yo where yo, yo ER dy(t) (a) Recall the vector space (C' [0, 27], R) from homework 2: Coo [a, b ] = { f : [a, b] -+ R | din f(x) continuous for all n 2 0 Prove that V C Co [0, 27] and then prove V is a subspace of (Co [0, 27], IR). (b) Show that cos(t) E V and sin(t) E V. (c) Using the following inner product 2 7 (y1 (t), yz (t) ) = y1(t )yz(t) dt prove that (cos(t), sin(t)) = 0. (d) Can B = {cos(t), sin(t) } be used as a basis for the vector space (V, R)? The following theorems and definitions below will help: ESE 551 Theorem 1. Suppose B is an orthogonal set of nonzero vectors. Then B is linearly independent. Theorem 2. Let V be an n-dimensional vector space and let B be a collection of vectors drawn from V. Then, B is a basis if and only if B contains n linearly independent vectors. Definition 3. A set B of vectors in V is called a basis for V if every vector v E V can be uniquely expressed as a finite linear combination of vectors in B