There are four simple linear transformations that can easily be described by multiplication of a 2 x 2 matrix. These types of matrices are used for many different applications, including in the computer graphics that you see in special effects at the movies.

The first is rotation.

Suppose that we want to find the 2 x 2 matrix that describes rotation of the diver by 90 degrees in the counterclockwise direction. Consider first the line connecting to .

After rotating this line by 90 degrees in the counterclockwise direction (about the point ) we should get the new line connecting to .

The 2 x 2 matrix that takes to and to is given by:

since

and

See what happens when we apply this transformation to every point on the diver.

More generally rotation of the line connecting to by degrees in the counterclockwise direction takes us to the new line connecting to . And rotation of the line connecting to by degrees in the counterclockwise direction takes us to the new line connecting to .

We can find the 2 x 2 transformation matrix as follows. We need

to be equal to , i.e. b = -sin and d = cos and we need

to be equal to , i.e. a = cos and c = sin .

Thus, the rotation by degrees in the counterclockwise direction about the point on the plane is given the transformation matrix:

Rotations

Try out various choices of q to see the rectangular diver rotate about the origin.

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This window shows the side view of a diver. The diver can be rotated about the origin by entering the value of theta in the appropriate textfield and then by pressing "Transform". To return the diver to the original orientation press "Reset". The coordinates of a point on the graph can be obtained by clicking anywhere on the graph. The x and y coordinates will be displayed in the lower left hand side of the applet. To zoom in or zoom out, click the appropriate button.