Solved: Assume the following trigonometric identities (see

Chapter 4, Problem 34E

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Assume the following trigonometric identities (see Exercise 37 in Section 4.1). Let H be the subspace of functions spanned by the functions in . Then is a basis for H, by Exercise 38 in Section 4.3.a. Write the -coordinate vectors of the vectors in C, and use them to show that C is a linearly independent set in H.b. Explain why C is a basis for H.Reference 37 in Section 4.1:[M] The vector space contains at least two interesting functions that will be used in a later exercise: Study the graph of f for , and guess a simple formula for f (t). Verify your conjecture by graphing the difference between 1 + f (t) and your formula for f (t) (Hopefully, you will see the constant function 1.) Repeat for g.Reference 38 in Section 4.3:[M] Show that is a linearly independentset of functions defined on . Use the method of Exercise 37. (This result will be needed in Exercise 34 in Section 4.5.)