Linear Transformations

Higher Level Leaving Cert Ireland

Here are some links which use applets to illustrate the ideas of Linear Transformations

1. http://www.ies.co.jp/math/java/misc/lintra/lintra.html Input the elements of the 2*2-matrix, then click "Transform" button. Experiment with the matrix in order to find transformations which are Reflections in the x-axis; Reflections in the y-axis; Central Symmetry in the Origin; Rotations about the Origin (use Trigonometric values from Sin and Cos of angles for the matrix input). Remember that the transformation has the form

where the matrix

(See below for other matrices).

2. http://www.math.hawaii.edu/phototranspage.html Transform your photos! courtesy of http://www.math.hawaii.edu/~ralph in Hawaii. This is a really clever use of Linear Transformations.

3. http://www.poisonapplet.com Leaving Cert by Trinity College students and http://www.poisonapplet.com/vectors/linear_transformations This site has much more on various aspects of the Leaving Cert for Higher Math Students. Requires a basic understanding of what is going on but really enhances understanding.

4. http://www.math.wisc.edu/~robbin/Transform/Transform.html Do a drawing which is ‘imitated’ by a transformation

A quick way of seeing the effect of a transformation on your own creations. The transformations used are the following:

The transformation has the form

where the matrix

is one of the following:

• Rotation through t degrees:

(Buttons change t)

• Rescale of the x coordinate:

(Buttons change c)

• Rescale of the y coordinate:

(Buttons change c)

• Interchange of the x-axis and y-axis:

(Buttons do nothing)

• Shear along the x axis:

(Buttons change c)

• Shear along the y axis:

(Buttons change c)

• Produces random values for the coefficients a, b, c, d. (In this case the buttons produce a new random matrix of coefficients.)

Page compiled by Neil Hallinan ©2003