Applied Linear Algebra

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MAT 350 Module Three Application Project Guidelines and Rubric: 2-D Graphic Transformations

This project gives a brief introduction to 2-D computer graphics. Students will learn how collections of points in can be visualized and how different linear transformations applied to these points manipulate and alter the original shape. Your answers to the problems should be typed into a single Word document. You should show your work for each part of each problem. Figures you create for this assignment can be hand-drawn and scanned, or you can sketch them in PowerPoint, Paint, or any other software tool of your choice. Visualizing Linear Transformations of the Plane provides general background theory on this topic and also works similar examples to those you are asked to work below.

Problems

1. Find a 2 x 2 matrix that maps e1 to –e2 and e2 to e1+3e2.

2. Consider the linear transformation described by the matrix. Note that arbitrary points on the x-axis can be written as for and arbitrary points on the y-axis can be written as for . To what set of points does this transformation map the x-axis to? To what set of points does this transformation map the y-axis to?

3. Define the unit square as the collection of points in with vertices (0,0), (0,1), (1,0), and (1,1). Sketch the unit square.

4. For each of the following linear transformation matrices, sketch the transformation of the unit square.

5. Find a transformation matrix that maps the unit square to the parallelogram with vertices (0,0), (2,4), (1,3), and (3,7).

Rubric

Critical Elements	Exemplary (100%)	Proficient (85%)	Needs Improvement (55%)	Not Evident (0%)	Value
Main Elements	Completes all of the requirements of all assigned problems	Completes most of the requirements	Includes some of the requirements	Includes fewer than half of the requirements; completes fewer than half of the problems	25
Computational Result (Correct Answers)	All parts of the assignment are computed correctly (all answers correct)	Most parts of the assignment are computed correctly	Some parts of the assignment are computed correctly	There are many computational errors	20
Computational Analysis (Shows Work)	Substantially shows work in a meaningful sequence to support answers	Shows work in an ordered sequence to support answers	There is some work shown, but it is neither comprehensive nor in an ordered sequence	Little or no work shown to support answers, or it is unclear how computations link to correct answers	30
Applies Course Content	All of the relevant course content is correctly applied to the assignment	Most of the relevant course content is correctly applied to the assignment	Some of the relevant course content is correctly applied to this assignment	Does not correctly apply any of the course content	15
Quantitative Literacy	Effectively works with linear algebra and solves the quantitative problems in the context of two-dimension computer graphics	Works with linear algebra and solves the quantitative problems in the context of two-dimension computer graphics, but with moderate effectiveness	Uses linear algebra and solves the quantitative problems with minimal effectiveness	Is not able to connect linear algebra techniques to two-dimension computer graphics	10
Earned Total	100%