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Math 111 Assignment 5 Due Tuesday Nov. 17, 2016 1. The linear transformation T: R 2 R 2 is pictured at the right.
The starting box is a square with sides x=?1 and y=?1. The
image is a parallelogram with corners at (?2, 3/2), (0, 3/2),
(2, ?3/2), (0, ?3/2) as illustrated at the right.
(a) Find the matrix of T in two different ways (and check that
you get the same result).
1 0 (i) Find the images of and directly from the diagram.
0 1 (ii) Write T as the composition of, first a horizontal shear S, then a dilation D, then a reflection R; find the
matrix of each, and multiply them together. That is, I want T of the form: T = RDS.
(b) Let Q be the transformation that applies T twice, that is, it maps x to T(T(x)). Find the matrix of Q.
Check your answer by using the diagram to track the path of the top left-hand corner of the starting box, 1
and compare with what you get by applying the matrix to .
(c) Find the matrix of the transformation Z that ?inverts? T, that is, that maps the parallelogram on the
right to the square on the left. Again, check your answer by following a suitable point. 2. The transformation T that maps every point in R3 into its reflection in the plane x + y ? 2z = 0 is
linear. Find its 3?3 matrix. As a check take a point on the plane and see if T maps it to itself. 3. In the diagram below, the 2?2 shaded square centered at the origin is bounded by the lines x = ?1 and
y = ?1. Let T denote the linear transformation that maps the square on the left to the parallelogram at the
right. T (a) Use the structure theorem for linear transformations to find the matrix of T. Check your work with a
suitable point, e.g. x = [1, 1],
(b) Now write T as a product of three transformations, first a dilation D, second a horizontal shear S, and
third a rotation R about the origin. Find matrices for each of the three transformations, and check that
their product RSD gives you the matrix of T. math 111 assignment 4 fall 2015 1 (c) Using the template below, draw the image of the transformed square after each of the three steps.
Take care with the accuracy of your drawing. T D R S math 111 assignment 4 fall 2015 2
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