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Question 1.12.2: Consider the set H of all functions of the form y = c1 sin t...

Consider the set H of all functions of the form y = c_{1} sin t +c_{2} cos t . In the vector space C of all continuous functions, explain why the set B = {sin t , cos t } is a basis for the subspace H.

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First, we recall that H is indeed a subspace of C[−1,1] due to our work in example 1.11.5.

By the definition of H (the set of all functions of the form y = c_{1} sin t +c_{2} cos t ), we see immediately that B is a spanning set for H. In addition, it is clear that the functions sin t and cos t are not scalar multiples of one another: any scalar multiple of sin t is simply a vertical stretch of the function, which cannot result in cos t . This tells us that the set B = {sin t , cos t } is also linearly independent, and therefore is a basis for H.

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