Question Use Matlab (4) This is a problem about trigonometric interpolation and you will need to solve a linear system \( A x=b \), where \( A \in \mathbb{R}^{n \times n} \) and \( b \in \mathbb{R}^{n} \). For the purpose of this problem, use the "backslash" operator in Matlab to solve the system. That is: \( \quad \mathrm{x}=\mathrm{A} \backslash \mathrm{b} \). You (just as Gauss did, circa 1803) have the following table of observations relating the ascension \( t \), in degrees, and the declination \( y \) (in minutes) of the asteroid Pallas (the second to be discovered, after Ceres, and presently the 3rd largest one): We want to interpolate these data with the function (0.1) \( f(t)=a_{0}+\sum_{k=1}^{5}\left[a_{k} \cos (2 \pi k t / 360)+b_{k} \sin (2 \pi k t / 360)\right]+a_{6} \cos (12 \pi t / 360) \), so that the uknowns are \( a_{0}, a_{1}, \ldots, a_{6} \) and \( b_{1}, \ldots, b_{5} \). This is a classical interpolation problem, and it has a unique solution. Now, what is your task? (a) (15 points.) Set up and solve \( { }^{1} \) the linear system resulting from the requirement \( f\left(t_{i}\right)=y_{i}, i=1,2, \ldots, 12 \). (b) (15 points.) Plot the original data points along with the curve obtained by using the function \( f(t) \) in (0.1) with the parameter values found in part (a), by sampling \( f(t) \) at 120 equispaced points in \( t,[0,3,6, \ldots, 357] \), and using that \( f(360)=f(0) \).

【General guidance】The answer provided below has been developed in a clear step by step manner.Step1/2Here's the MATLAB code for parts (a) and (b) separately:Part (a)% Given observationst = [0,30,60,90,120,150,180,210,240,270,300,330];y = [408,89,-66,10,338,807,1238,1511,1583,1462,1183,804];% Construct matrix An = length(t); % Number of observationsA = zeros(n, 12); % Initialize matrix AA(:, 1) = 1; % Set first column of A to 1for k = 1:5 A(:, 2*k) = cos(2*pi*k*t/360); A(:, 2*k+1) = sin(2*pi*k*t/360);endA(:, 12) = cos(12*pi*t/360);% Solve the linear system Ax = yx = A \ y';% Display the solutiondisp('Coefficients:');disp(x'); The code above corresponds to part (a) of the problem, where we need to set up and solve the linear system \( \mathrm{{A}{x}={y}} \) to find the coefficients \( \mathrm{{a}_{{0}},{a}_{{1}},\ldots,{a}_{{6}}} \) and \( \mathrm{{b}_{{1}},\ldots,{b}_{{5}}} \) that interpolate the given observations. Here's an explanation of the code:We first define the given observations as two vectors, t and y.We then initialize the matrix A with zeros, and set the first column to 1 (for the constant term in the trigonometric function).We then loop over the indices k from 1 to 5, and compute the columns of A corresponding to the cosine and sine terms of the trigonometric function. We use the cos and sin functions, along with the values in the t vector, to compute these columns.Finally, we compute the column of A corresponding to the cosine term with frequency 12, and solve the linear system using the backslash operator in Matlab (x = A \ y'). The resulting vector x contains the coefficients of the trigonometric function that interpolate the observ ... See the full answer