2.2 linear transformations in geometry for an animation of this topic visit
TRANSCRIPT
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?
We should have these basic matrices in our library
Identity Matrix
Rotations
Scaling
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
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.
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:
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?
Identity Matrix
Projection Matrices
Projection onto x-axis
Projection onto y-axis
Rotation Matrix
One directional ScalingMixed ScalingHorizontal ShearVertical ShearScaling