This assignment is designed to show you two important things: that matrices when they act upon...

 

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This assignment is designed to show you two important things: that matrices when they act upon a vector cause them to move or change length and that the order in which you apply more than one matrix to a vector changes what happens to that vector. 1. First generate at least 11 vectors of unit length that radiate out from the origin spaced equal angles apart (think of this as each vector pointing to one of the numbers on a clock face). Plot these vectors using different colours or otherwise label them so you can keep track of them. 2. Then compose several 2x2 matrices to do the following operations: rotate your vectors by an angle (choose your own value of 0). In this case the matrix is this one R- [cos-sin sin cos 0 reflect your vectors across the x axis. reflect your vectors across the y axis. stretch your vectors in both the z and y direction. You choose your scaling factors. I strongly suggest you make them different and don't choose a scaling factor of 1. We already know that multiplying a vector by the identity matrix does not change the vector! Experiment with this by multiplying your matrices by your vectors and plotting the results. . For the report, include a plot of your original vectors and a plot of what happens to them under each of the above operations. Be smart about how you choose your plots. 3. Next choose 2 different geometrical operations (let's call those matrices A and B) and do the following. First compute ABu for all of your vectors and plot that. Then compute B Av for all of your vectors and plot that on a separate graph. Experiment with the choices of A and B until you find plots where the graph of ABBA for all of your vectors (that is, find two geometric transformations that commute). Include this plot in your report and clearly identify the matrices A and B and write out in plain English what is occuring. Then find matrices where the graph of ABU BAu for all of your vectors. Include this plot in your report and clearly identify the matrices A and B for this case. This assignment is designed to show you two important things: that matrices when they act upon a vector cause them to move or change length and that the order in which you apply more than one matrix to a vector changes what happens to that vector. 1. First generate at least 11 vectors of unit length that radiate out from the origin spaced equal angles apart (think of this as each vector pointing to one of the numbers on a clock face). Plot these vectors using different colours or otherwise label them so you can keep track of them. 2. Then compose several 2x2 matrices to do the following operations: rotate your vectors by an angle (choose your own value of 0). In this case the matrix is this one R- [cos-sin sin cos 0 reflect your vectors across the x axis. reflect your vectors across the y axis. stretch your vectors in both the z and y direction. You choose your scaling factors. I strongly suggest you make them different and don't choose a scaling factor of 1. We already know that multiplying a vector by the identity matrix does not change the vector! Experiment with this by multiplying your matrices by your vectors and plotting the results. . For the report, include a plot of your original vectors and a plot of what happens to them under each of the above operations. Be smart about how you choose your plots. 3. Next choose 2 different geometrical operations (let's call those matrices A and B) and do the following. First compute ABu for all of your vectors and plot that. Then compute B Av for all of your vectors and plot that on a separate graph. Experiment with the choices of A and B until you find plots where the graph of ABBA for all of your vectors (that is, find two geometric transformations that commute). Include this plot in your report and clearly identify the matrices A and B and write out in plain English what is occuring. Then find matrices where the graph of ABU BAu for all of your vectors. Include this plot in your report and clearly identify the matrices A and B for this case.

Answer rating: 100% (QA)

To complete the assignment lets follow the steps provided 1 Generate Vectors To generate 11 vectors radiating out from the origin spaced equally apart we can use the angles from 0 to 2 360 degrees and ... View the full answer