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In order to guarantee that management fully understands the process, you have mapped out the following steps to ensure you have captured the process and have data to support your findings and to share with management. Your plan is to demonstrate computations on a simple $33$ matrix where the computations are easier to follow. Then you will perform similar computations on a large image to compress the image data without significantly degrading image quality.

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.

Consider the matrix: $33$ :

$A=\left[\begin{array}{ccc}1& 2& 3\\ 3& 3& 4\\ 5& 6& 7\end{array}\right]$

Use the svd$()$ function in MATLAB to compute $A_1,$ the rank$-1$ approximation of $A.$ Clearly state what $A_1$ is$,$ rounded to $4$ decimal places. Also, compute the root mean square error $($RMSE$)$ between A and $A_1.$

$2.$ Use the svd$()$ function in MATLAB to compute $A_2,$ the rank$-2$ approximation of $A.$ Clearly state what $A_2$ is$,$ rounded to $4$ decimal places. Also, compute the root mean square error $($RMSE$)$ between A and $A_2.$ Which approximation is better, $A_1$ or $A_2?$ Explain.

$3.$ For the $33$ matrix $A,$ the singular value decomposition is $A=US{V}^{\text{'}}$ where $U=[u_1{u}_2{u}_3].$ Use MATLAB to compute the dot product $d_1=$dot$(u_1,u_2).$ Also, use MATLAB to compute the cross product $c=$cross$(u_1,u_2)$ and $dotproduct{d}_2=$dot$(c,u_3).$ Clearly state the values for each of these computations. Do these values make sense? Explain.

$4.$ Using the matrix $U=[u_1{u}_2{u}_3],$ determine whether or not the columns of $U$ span $R^3.$ 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 $k$ that will result in a compression ratio of $CR$~~$2.$ 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.

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