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 prod$c=$cross$(u_1,u_2)$ and dot product $d_2=$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.

What to Submit

To complete this project, you must submit the following:

Use the provided Project Two Template as the starting point for your project solution. Complete each portion of the template, run the project, and then export your work as a single PDF file. Upload this PDF document that shows your answers and supporting work for the problems described above. Make sure to include explanations of your work, as well as all MATLAB code and outputs of the computations.

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