2.2 Linear Transformations in Geometry For an animation of this topic visit

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

Answer: 3I

Transformation matrices Use 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 matrices How would the following matrices transform that L? (May check via website listed on initial slide) Scale by factor of 2 Projection onto Horizontal 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: u 1 and u 2 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 scalingCombined scaling What would a single component scaling or combined scaling matrix look like in R n ?

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 Scaling Mixed Scaling Horizontal Shear Vertical Shear Scaling

Homework: p , 8-10, 26 a-c only,30,31

Rotations