2114-Homework3.pdf

2114-Homework3.pdf

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Math 2114 Transformations Homework 3 Name: 1. Let A = 3 0 0 - 2 and define T : R 2 R 2 by T ( ~x ) = A~x . Find the images under T of ~u = (3 , - 1) and ~v = (0 , 1 . 5) and the vector ~u + ~v . Also sketch the three vectors before and after the transformation. Write 1–2 sentences that explain how you found your answers/knew they were right and why the sketch makes sense. 2. Let A = 1 - 5 - 7 - 3 7 5 and define T : R 3 R 2 by T ( ~x ) = A~x . (a) Find the image under T of ~u = (2 , 1 , - 1). (b) Find a vector ~x whose image under T is ~ b = ( - 12 , 12). Explain why your work makes sense. 3. Show that the transformation T defined by T x 1 x 2 = x 1 + 1 x 2 is not a linear transformation. 4. After class, two linear algebra students start talking about linear transformations and the letter “N.” One of the students suggested translation (shifting up) as another linear transformation that could be done to the letter “N,” like the following: The other student disagreed, stating that shifting the “N” up like this is NOT an example of a linear transformation. Which student is right? Why?

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