To develop your idea proposal, work the problems described below. As you complete each part, make sure to show your work and carefully describe how you arrive at your final answer. Any MATLAB code or MATLAB terminal outputs you generate should be included in your idea proposal to support your answers and work. 1. Consider the matrix: 3 × 3: [1 2 31 A = 3 34 [567] Use the svd() function in MATLAB to compute A₁, the rank-1 approximation of A. Clearly state what A₁ is, rounded to 4 decimal places. Also, compute the root mean square error (RMSE) between A and A₁. 2. Use the svd() function in MATLAB to compute A2, the rank-2 approximation of A. Clearly state what A₂ is, rounded to 4 decimal places. Also, compute the root mean square error (RMSE) between A and A₂. Which approximation is better, A₁ or A₂? Explain. 3. For the 3 x 3 matrix A. the singular value decomposition is A = USV' where ʊ = [u₁₂u] Use MATLAB to compute the dot product d₁ = dot(u₁, u₂). Also, use MATLAB to compute the cross product c = cross(u₁, u2) and dot product d₂ = dot(c, us). Clearly state the values for each of these computations. Do these values make sense? Explain. 4. Using the matrixʊ = [u₁ u2 u3], determine whether or not the columns of U span R³. Explain your approach. 5. Use the MATLAB imshow() function to load and display the image A stored in the provided MATLAB image.mat file (available in the Supporting Materials area). For the loaded image, derive the value of that will result in a compression ratio of CR2. For this value of k construct the rank-k; approximation of the image. 6. Display the image and compute the root mean square error (RMSE) between the approximation and the original image. Make sure to include a copy of the approximate image in your report. 7. Repeat steps 5 and 6 for CR 10, CR 25, and CR 75. Explain what trends you observe in the image approximation as CR increases and provide your recommendation for the best CR based on your observations. Make sure to include a copy of the approximate images in your report.