Linear transformations of the plane - Harvard University

MATH21B ? LECTURE 5: TRANSFORMATIONS IN GEOMETRY SPRING 2018, HARVARD UNIVERSITY

1. Linear transformations of the plane

Problem 1. Describe in words the effect on the plane of the linear transformation corresponding to the following matrices:

10 (a)

0 -1

20 (b)

02

1 -1 (c)

01

0 -1 (d)

10

00

(e)

.

01

Solution. (a) A reflection in the x-axis. (b) A dilation by a factor of 2. (c) A horizontal shear. (d) A rotation by 90 counter-clockwise. (e) A projection onto the y-axis.

Problem 2. (i) Find the matrix A for the linear transformation T : R2 R2 given by reflection in the line y = x.

(ii) What would be the matrix B be for the linear transformation obtained by reflection in the line y = x and then dilating by a factor of 3?

(iii) Is A invertible? If so, what is its inverse?

Solution.

(i) Drawing a picture, we see that T swaps e1 and e2. Thus the matrix A for T is given by 0 1. 10

(ii) Drawing a picture, we see that it sends e1 to 3e2 and e2 to 3e1. Thus the matrix B is 03 . 30

(iii) Yes, because we can see geometrically that T 2 fixes all vectors. Thus A-1 = A.

2. Auto-straightening

Problem 3 (Challenge). In Figure 1 you see a photo I took, and my phone's attempt to automatically straighten it. The ratio of the edges is 3 : 4, the straightening angle is 9.7, and photo is slightly enlarged so that the original image exactly fits inside the rotated image (so as little of the picture as possible is lost). Give the linear transformation that is applied to the picture.

Solution. The linear transformation is a rotation with dilation. We know the angle is 9.7 and the direction is counter-clockwise, so the matrix will be of the form

r cos(2 ? 9.7/360) -r sin(2 ? 9.7/360) . r sin(2 ? 9.7/360) r cos(2 ? 9.7/360)

Let us shorten 2 ? 9.7/360 to . The difficulty is figuring out the scaling factor r. Consider figure 2. From

it we conclude that

and

we

can

solve

r

=

3+4 sin() 3 cos()

1.242.

3 + 4 sin() cos() =

3r

1

2

21B

Figure 1. Auto-straightening.

3r

4 sin()

3

4

Figure 2. Some observations.

3. Self-test

Problem 4. (i) True or false: an invertible matrix has to be square. (ii) Give an example of (2 ? 2)-matrix whose linear transformation acts on the plane R2 as a shear. (iii) True or false: The (2 ? 2)-matrix corresponding to a rotation of the plane always has rank 2. (iv) Give an example of (3 ? 3)-matrix whose linear transformation acts on three-dimensional space R3 by reflection in a plane. (v) Explain why a transformation T : R3 R3 satisfying T (e1) = e1 and T (2e1) = e2 can not be linear.

Solution. (i) True, as every equation Ax = b has to have a unique solution and thus rref(A) has to be square with 1's on the diagonal.

(ii) See 1(c). (iii) True, it is invertible with inverse rotation by the same angle in the other direction.

21B

3

(iv) The following matrix is reflection in the yz-plane:

-1 0 0 0 1 0 .

0 01

(v) It doesn't satisfy T (x) = T (x) for = 2 and x = e1.

Summary

? To see what a linear transformation does look at the columns. ? Particular useful linear transformations to remember are: rotations, dilations, reflections, and

projections. Of these the first three are invertible, but projections are not.

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download