Learning Goal: I'm working on a linear algebra project and need an explanation and answer to help me learn.Project Two Guidelines and RubricCompetencyIn this project, you will demonstrate your mastery of the following competencies:Interpret the properties of vector spaces
Utilize advanced matrix techniques to solve complex linear problems
ScenarioYou are employed as a computer programmer for a popular social media site that stores a large amount of user media files. You believe you have found a way to reduce costs by compressing image files using singular-value decomposition (SVD). The compressed files would require less storage space, which would result in savings for the company. You think it will work, but you want to test your theory and record the steps you take to use as a reference when sharing your idea with management.DirectionsIn 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 3 x 3 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: :Use the svd() function in MATLAB to compute , the rank-1 approximation of . Clearly state what is, rounded to 4 decimal places. Also, compute the root mean square error (RMSE) between and .
Use the svd() function in MATLAB to compute , the rank-2 approximation of . Clearly state what is, rounded to 4 decimal places. Also, compute the root mean square error (RMSE) between and . Which approximation is better, or ? Explain.
For the matrix , the singular value decomposition is where . Use MATLAB to compute the dot product . Also, use MATLAB to compute the cross product and dot product . Clearly state the values for each of these computations. Do these values make sense? Explain.
Using the matrix , determine whether or not the columns of span . Explain your approach.
Use the MATLAB imshow() function to load and display the image 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 . For this value of , construct the rank- approximation of the image.
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.
Repeat steps 5 and 6 for , , and . Explain what trends you observe in the image approximation as increases and provide your recommendation for the best based on your observations. Make sure to include a copy of the approximate images in your report.
What to SubmitTo 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.Supporting MaterialsThe following resource(s) may help support your work on the project:SVD and Image Compression OverviewProject Two TemplateThis template is the starting point for your solution to this project. Download the template, open in MATLAB, and complete each portion. In addition to this, you will need to download and utilize the MATLAB image.mat file.Project Two RubricCriteriaExemplaryProficientNeeds ImprovementNot EvidentValueRank-1 ApproximationN/AUses the SVD function to compute the rank-1 approximation (100%)Shows progress toward proficiency, but with errors or omissions; areas for improvement may include constructing an accurate matrix (55%)Does not attempt criterion (0%)10Rank-2 ApproximationN/AUses the SVD function to compute the rank-2 approximation (100%)Shows progress toward proficiency, but with errors or omissions; areas for improvement may include constructing an accurate matrix (55%)Does not attempt criterion (0%)10Dot and Cross Product ComputationExceeds proficiency in an exceptionally clear, insightful, sophisticated, or creative manner (100%)Interprets the results of dot and cross product computation (85%)Shows progress toward proficiency, but with errors or omissions; areas for improvement may include focusing on the accuracy of computation and interpretation of results (55%)Does not attempt criterion (0%)10SpanExceeds proficiency in an exceptionally clear, insightful, sophisticated, or creative manner (100%)Explains the approach for determining whether the columns span the given space (85%)Shows progress toward proficiency, but with errors or omissions; areas for improvement may include using a valid approach for determining span (55%)Does not attempt criterion (0%)15Compression RatioN/AComputes the values of k that will result in the specified compression ratios (100%)Shows progress toward proficiency, but with errors or omissions; areas for improvement may include focusing on the accuracy of the values of k (55%)Does not attempt criterion (0%)10Rank ApproximationN/AConstructs the rank-k approximation of the image based on the specified value of k (100%)Shows progress toward proficiency, but with errors or omissions; areas for improvement may include constructing the images using the required steps and operations (55%)Does not attempt criterion (0%)10RMSEN/AComputes the RMSE between the approximations and the original image (100%)Shows progress toward proficiency, but with errors or omissions; areas for improvement may include computing the RMSE for all of the approximations (55%Does not attempt criterion (0%)10Impact of Increasing CRExceeds proficiency in an exceptionally clear, insightful, sophisticated, or creative manner (100%)Explains trends observed in the image approximations as CR increases (85%)Shows progress toward proficiency, but with errors or omissions; areas for improvement may include accurately interpreting the data observed (55%)Does not attempt criterion (0%)15Articulation of ResponseExceeds proficiency in an exceptionally clear, insightful, sophisticated, or creative manner (100%)Clearly conveys meaning with correct grammar, sentence structure, and spelling, demonstrating an understanding of audience and purpose (85%)Shows progress toward proficiency, but with errors in grammar, sentence structure, and spelling, negatively impacting readability (55%)Submission has critical errors in grammar, sentence structure, and spelling, preventing understanding of ideas (0%)10Total:100%
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