answer question 2 and 3 only

(Least Squares Curve Fitting) Consider the function of the form y = Bi sin(x) + 82 sin(2x) + 83 sin(3x) + Basin(4x) + 85 sin(51) + and the points (a) [7 marks] If the function exactly went through these 10 points, what linear system would the B1, B2, 83, BA, Bs satisfy? (write as a matrix vector equation) (b) [7 marks] What is corresponding system the least squares solution satisfies (i.e. the normal equations)? (c) [6 marks] Find the least squares solution. 2. (Fourier Series) Piecewise continuous bounded functions can also be expressed as Fourier Series. Let J(I) = 1, for I an cos(nz) + b, sin(nz) n=1 n=1 on the interval [0, 2x] (b) [5 marks] Plot J(r) and an + En=1 an cos(nr) + En=1 b,, sin(nr) on the interval [0, 27] (and on the same graph). (c) [5 BONUS marks] Find a closed form formula for an and b,, in terms of n. 3. (Discrete Fourier Transform) (a) [5 marks] Divide the interval [0, 27] into 10 equal pieces and sample the function J(x) (from question 2) at the left-hand endpoints to get a vector vy in 10. (b) [10 marks] Compute the Discrete Fourier Transform of the vector vJ. (c) [5 marks] Discuss the commonalities between your answers to question 1(c), question 2(a) and question 3(b) and why you think these commonalities occur