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Homework: Using Matrices for Transformation

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides practice problems related to using matrices to perform transformations.

Exercise 1

Harpoona’s best friend is a fish named Sam, whose initial position is represented by the matrix:

[ S 1 ] = [ 0 4 4 0 0 4 0 0 3 3 0 3 ] [ S 1 ]=[ 0 4 4 0 0 4 0 0 3 3 0 3 ]

Draw Sam:

An empty graph for plotting.

Exercise 2

When the matrix T = 1 2 [ 3 -1 1 3 ] T= 1 2 [ 3 -1 1 3 ] is multiplied by any matrix, it effects a powerful transformation on that matrix. Below, write the matrix S 2 = T S 1 S 2 =T S 1 . (You may use 1.7 as an approximation for 33 size 12{ sqrt {3} } {}.)

Exercise 3

Draw Sam’s resulting condition, S 2 S 2 .

Exercise 4

The matrix T –1 T –1 will, of course, do the opposite of T T. Find T –1 T –1 . (You can use the formula for the inverse matrix that we derived in class, instead of starting from first principles. But make sure to first multiply the 1 2 1 2 into T T, so you know what the four elements are!)

Exercise 5

Sam now undergoes this transformation, so his new state is given by S 3 = T –1 S 2 S 3 = T –1 S 2 . Find S 3 S 3 and graph his new position.

Exercise 6

Finally, Sam goes through T –1 T –1 again, so his final position is S 4 = T –1 S 3 S 4 = T –1 S 3 . Find and graph his final position.

Exercise 7

Describe in words: what do the transformations T T and T –1 T –1 do, in general, to any shape?

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