Notes on using MatrixTrans.xls

This spreadsheet allows students to see the effect of a matrix transformation on the unit square. It can be used to discover and/or familiarise oneself with the standard transformation matrices for enlargements stretches shears reflections and rotations.

Background:

The idea of a linear transformation of the plane and it’s representation by a 2 by 2 matrix.

Multiplication of matrices and composition of transformations.

Some techniques for finding the matrix of a transformation (for instance by finding the images of  and which give the first and second columns of the matrix respectively).

Basic trigonometry and familiarity with the trig ratios of the special angles 30º, 45º,60º.

Questions for students.

When you discover something make a note of it so that you build up a table of results by the end of the exercise.

Some matrices may be the result of two more basic transformations combined together in which case the order is important.

• What transformation is represented by the matrix ?

• Find the matrix for an enlargement factor k.

• What transformation is represented by the matrix ?

• Find the matrix for a refection in the y-axis.

• What transformation is represented by the matrix ?

• Find the matrix for a rotation of 90º anticlockwise about the origin.

• Find the matrix for a rotation of 90º clockwise about the origin.

• What transformation is represented by the matrix ?

• Find the matrix for a stretch factor k parallel to the y-axis.

• What transformation is represented by the matrix ? (If you can’t see it try instead).

• What transformation is represented by the matrix ?

• What transformation is represented by the matrix ?

• A shear parallel to the x-axis factor k is a transformation where points on the x-axis stay put and other points move parallel to the x-axis by an amount equal to k times their y-coordinates. Find the matrix for this transformation.

•   Find the matrix for a shear parallel to the y-axis factor k.

• Find the matrix for a rotation of 30º anticlockwise about the origin.

• Find the matrix for a rotation of 60º clockwise about the origin.

• Find the equation for the straight line through the origin which makes an angle of 60º with the positive x-axis. Find the matrix for a reflection in this line.

• Find the matrix for a rotation of θº anticlockwise about the origin.

•   Find the matrix for a reflection in the straight line through the origin which makes an angle of θº with the positive x-axis.

•   Find the matrix for a stretch parallel to the y-axis factor 2 followed by rotation of 30º clockwise about the origin.

• What transformation is represented by the matrix ?

• What transformation is represented by the matrix ?