2.2 Linear Transformations in Geometry

For an animation of this topic visit

http://www.ies.co.jp/math/java/misc/don_trans/don_trans.html

Library of basic matrices

What matrices do we have in our library of basic matrices?

Library of basic matrices

What matrices do we have in our library of basic matrices?

We should have these basic matrices in our library

Identity Matrix

Rotations

Scaling

Problem 32

Problem 32

Answer: 3I

Transformation matricesUse your knowledge of matrix multiplication

(and your library of matrices) to predict what affect these matrices would have on our dog.

How would the following matrices transform that L? (May check via website listed on initial slide)

Transformation matricesHow would the following matrices transform

that L? (May check via website listed on initial slide)

Scale by factor of 2 Projection ontoHorizontal axis

Reflect about vertical axis (y-axis)

Add these (the last two) to your list of library of basic matrices.Find a matrix that describes a projection onto the y-axis and add it to your library of matrices.

What type of Linear Transformation results from these matrices

(Answer on next slide)

What type of Linear Transformation results from these matrices

Reflect about Horizontal Shear rotated 45 degrees

Horizontal axis and scaled by root 2

Add the first one to your library of basic matrices. We will generalize the last two before adding them.

What do you think that these matrices would do to our dog?

Horizontal and vertical shear

This leaves one component unchanged while skewing the points in the other direction

Horizontal shear Vertical shear

Here is an example of horizontal shear

Recall: Scaling

For any positive constant k, the matrix

Defines a scaling by k times. If k is between 0 and 1 then the scaling is a contraction. If k >1 then the scaling is a dilation (enlargement)

Projections

Consider a line L in the coordinate plane, running through the origin. Any vector in _ can be written as + =

The transformation T(x) = is called the projection onto x

Projections

Note: u1 and u2 are the components of a unit vector

This matrix is called a projection matrix. You will need it in your notes add this to your library of matrices

MV calc we know:

Example 2

Find the matrix A of the projection onto the Line spanned by

Example 2 Solution

From your knowledge of matrix multiplication what would these

matrices do to our dog?

One directional scaling(Note this is not in our text book)

These matrices multiply one component of b while leaving the other unchanged.

For example

Notice that the x components are halved while the y is unchanged

Combined scaling

This will multiply the x component by r and the y component by s

Add these to our library of basic matrices

Horizontal scaling Vertical scaling Combined scaling

What would a single component scaling or combined scaling matrix look like in Rn?

What matrices should we have in our library of basic matrices?

What matrices should we have in our library of basic matrices?

Identity Matrix

Projection Matrices

Projection onto x-axis

Projection onto y-axis

Rotation Matrix

One directional ScalingMixed ScalingHorizontal ShearVertical ShearScaling

Homework: p. 65 1-6, 8-10, 26 a-c only,30,31

Rotations