Let E = { 1, cost, cos2t , cos3t , cos4t , cos5t , cos6t } and F = { 1, cost, cos2t, cos3t, cos4t, cos5t, cos6t}

be two ordered bases for the vector space of continuous functions.

(a) Find [1]E, [cost]E , [cos2t]E , [cos3t]E , [cos4t]E , [cos5t]E , [cos6t]E .

(b) Use part (a) to show that F is a basis for S, the subspace of functions spanned by { 1, cost, cos2t , cos3t , cos4t , cos5t , cos6t }.

(c) Find the transition matrix T from basis E to basis F. (you can use your calculator)

(d) Recall from calculus that integrals such as

are tedious to compute. (you need to apply integration by parts repeatedly and use the half angle formula). Instead, use the transition matrix T or T-1 (which ever is appropriate) to compute the integral in an easier form.

be two ordered bases for the vector space of continuous functions.

(a) Find [1]E, [cost]E , [cos2t]E , [cos3t]E , [cos4t]E , [cos5t]E , [cos6t]E .

(b) Use part (a) to show that F is a basis for S, the subspace of functions spanned by { 1, cost, cos2t , cos3t , cos4t , cos5t , cos6t }.

(c) Find the transition matrix T from basis E to basis F. (you can use your calculator)

(d) Recall from calculus that integrals such as

are tedious to compute. (you need to apply integration by parts repeatedly and use the half angle formula). Instead, use the transition matrix T or T-1 (which ever is appropriate) to compute the integral in an easier form.

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