Let B = {1, cost, cos 2 t,..., cos 6 t and C = {1, cos t,

Question:

Let B = {1, cost, cos2 t,..., cos6 t and C = {1, cos t, cos 2t,...,cos 6t}. Assume the following trigonometric identities (see Exercise 45 in Section 4.1).

Let H be the subspace of functions spanned by the functions in B. 

a. Write the B-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.


Data from in Exercise 45

The vector space H = Span {1, cos2 t, cos4 t, cos6 t contains at least two interesting functions that will be used in a later exercise:

Study the graph of f for 0 ≤ t ≤ 2π, 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). (Hope- fully, you will see the constant function 1.) Repeat for g.

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Linear Algebra And Its Applications

ISBN: 9781292351216

6th Global Edition

Authors: David Lay, Steven Lay, Judi McDonald

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