TARUN GEHLOTS

Introduction to transformations

* Matrices can be used to represent many transformations on a grid (such as reflections, rotations, enlargements, stretches and shears). * To find the image of a point P, you multiply the matrix by the position vector of the point.

Example: A transformation is represented by the 2 by 2 matrix M = .
To find the image of the point (3, 2) under this transformation, you need to find the result of the following matrix multiplication

So the coordinates of the image are (8, -1).

Example 2: A rectangle has coordinates (1, 1), (4, 1), (4, 3) and (1, 3). Find the coordinates of the image of the rectangle under the transformation represented by the matrix .

Solution: You could find the image of each vertex in turn by finding , etc.
However, it is more efficient to multiply the transformation matrix by a rectangular matrix containing the coordinates of each vertex:
.
So the image has coordinates (2, 0), (11, -3), (9, -1) and (0, 2).

The diagram below shows the object and the image:

Any transformation that can be represented by a 2 by 2 matrix, , is called a linear transformation.

1.1 Transforming the unit square

The square with coordinates O(0, 0), I(1, 0), J(0, 1) and K(1, 1) is called the unit square.

Suppose we consider the image of this square under a general linear transformation as represented by the matrix : .

We therefore can notice the following things: * The origin O(0, 0) is mapped to itself; * The image of the point I(1, 0) is (a, c), i.e. the first column of the transformation matrix; * The image of the point J(0, 1) is (b, d), i.e. the second column of the transformation matrix; * The image of the point K(1, 1) is (a + b, c+ d), i.e. the result of finding the sum of the entries in each row of the matrix.

Example: Find the image of the unit square under the transformation represented by the matrix .

Solution:
The image of (1, 0) is (1, 0) (i.e. the first column)
The image of (0, 1) is (2, 1) (i.e. the second column)
The image of (1, 1) is (3, 1) (i.e. add the entries in the top row and the bottom row together).

We can show the unit square and its image in a diagram:

We notice that the points on the x-axis have not moved. This type of transformation is called a shear. Here the invariant line is the x-axis.

We can describe what transformation any matrix represents by seeing how it affects the unit square.

Example:
A transformation T is given by:
.
a) Find the image of the point A(3, 2).
b) Describe fully the transformation represented by T.

Solution:
a) The image of A(3, 2) can be found by:

So the image of A is the point (-2, 3).

Note: The image of a point A is often denoted .

b) To describe the transformation we consider the image of the unit square:
The image of (1, 0) is (0, 1) (i.e. the first column)
The image of (0, 1) is (-1, 0) (i.e. the second column)
The image of (1, 1) is (-1, 1) (i.e. add the entries in the top row and the bottom row together).

1.2 Finding the matrix to represent a transformation

To find the matrix that defines a transformation you find the images of the two points I(1, 0) and J(0, 1).
The image of (1, 0) forms the first column of the matrix.
The image of (0, 1) forms the second column of the matrix.

Example:
Find the matrix that represents a reflection in the y-axis.

Solution:
When you reflect in the y-axis: * the image of I(1, 0) is (-1, 0) * the image of J(0, 1) is (0, 1).

Therefore the matrix is:
.

Example 2:
Find the matrix that represents an enlargement centre (0, 0), scale factor 3.

Solution:
The image of the point I(1, 0) is (3, 0).
The image of the point J(0, 1) is (0, 3).

So the matrix is .

Example 3:
Find the matrix that represents a rotation centre (0, 0), 90 degrees clockwise.

Solution:
The image of the point I(1, 0) is (0, -1).
The image of the point J(0, 1) is (1, 0).

So the matrix is .

1.3 Some special matrix transformations

Rotation θ° anticlockwise centre (0, 0).
The matrix that represents this transformation is .
This matrix is given in the OCR formula book.

Reflection in the line y = (tanθ)x
The general form for the matrix corresponding to a reflection in the line y = (tanθ)x is
.
This matrix is also given in the OCR formula book.

Example:
Find the matrix of an anticlockwise rotation about the origin through 60°.

Solution:
This matrix would be
.

Example 2:
Find the matrix that corresponds to a reflection in the line y = 2x.

Solution:
Comparing the line y = 2x with the form y = (tanθ)x, we see that tanθ = 2 so that θ = 63.43°.
Therefore the matrix is:

.

Stretch, scale factor k parallel to the x-axis
The matrix for this transformation is .

Note: A stretch is an enlargement is one direction only.

Stretch, scale factor k parallel to the y-axis
The matrix for this transformation is .

Shear parallel to the x-axis
The matrix corresponds to a shear parallel to the x-axis. Points on the x-axis do not move, whilst points on the line y = 1 are translated k units to the right.

Shear parallel to the y-axis
The matrix corresponds to a shear parallel to the y-axis. Points on the y-axis do not move, whilst points on the line x = 1 are translated k units up.

The matrix M is given by .
(i) Give a complete geometrical description of the transformation represented by M.
(ii) Hence write down the smallest positive integer n for which .

Solution:
(i) When describing the transformation corresponding to a matrix, it is sensible to first compare the given matrix to the general matrices for rotation and reflection (as quoted in the formula book).
The matrix for rotating through θ° anticlockwise centre (0, 0) is

The matrix for reflecting in the line y = (tanθ)x is
.

Our matrix is similar in structure to the matrix for rotation (since the entries on the leading diagonal (from top left to bottom right) have the same signs and the entries on the other diagonal have opposite signs).

We therefore have to find the correct value for θ. We do this by solving and .
Since cosθ is negative and sinθ is positive, θ must be in the second quadrant (between 90° and 180°). Solving the equations, we find that θ = 120°.

So M represents a rotation centre (0, 0), through 120° in an anticlockwise direction.

(ii) Since M represents a rotation through one third of a turn, if you repeat this transformation 3 times you would end up where you started off. Therefore M3 would represent the identity transformation (which is represented by the identity matrix). So n = 3
The transformation T is represented by the matrix M, where
.
a) Give a geometrical interpretation of T.
b) Find the smallest positive value of n for which

The matrix M is .
a) Find (i) (ii) .
b) The transformation T is given by: .
Describe fully the geometrical transformation represented by T.

a) The transformation T is defined by the matrix A, where
.
(i) Describe the transformation T geometrically. (Hint: consider the unit square) (ii) Calculate the matrix product .
(iii) Explain why the transformation T followed by T is the identity transformation.
b) The matrix B is defined by
.
(i) Calculate B2 – A2. (ii) Calculate (B + A)(B – A).

Combining transformations

Suppose we have two matrices M and N, each corresponding to a different transformation.
Then the matrix product NM corresponds to the combined transformation, where M is the first transformation and N is the second transformation.

Note: Look carefully at the order.

Example: Find the matrix that represents the combined transformation: * reflection in the x-axis; * followed by enlargement scale factor 2 centre (0, 0).

Solution:
We first find the matrix that corresponds to each separate transformation.

Transformation 1: Refection in the x-axis The image of (1, 0) is (1, 0) The image of (0, 1) is (0, -1)
So the matrix is .

Transformation 2: Enlargement s.f. 2 centre (0, 0):
This matrix is .

Combined transformation: Since we wish to reflect first then enlarge, the correct order to multiply the matrices is: =

Worked examination question (AQA 2002)
A transformation T1 is represented by the matrix:
.
a) Give a geometrical description of T1.
The transformation T2 is a reflection in the line .
b) Find the matrix M2 which represents the transformation T2.
c) (i) Find the matrix representing the transformation T2 followed by T1. (ii) Give a geometrical description of this combined transformation.

Solution:
a) If you compare the matrix M1 with the general matrix for a rotation, i.e. , you will see that it corresponds to a rotation, centre (0, 0) through 30 degrees in anticlockwise direction.

b) The matrix for reflecting in the line y = (tanθ)x is
.
Here we want tanθ = √3, i.e. θ = 60°.
Therefore the matrix is .

c) (i) The matrix for T2 followed by T1 is =. (ii) This matrix product has the form of a reflection.
The equation of the mirror line is y = (tanθ)x where θ is found by solving the equations: and .
We find that 2θ = 150°, i.e. that θ = 75°.

So the combined transformation is equivalent to a reflection in the line y = (tan75)x.

a) The transformation T1 is defined by the matrix
.
Describe this transformation geometrically.
b) The transformation T2 is an anticlockwise rotation about the origin through an angle of 60°. Find the matrix of the transformation T2. Use surds in your answer where appropriate.
c) Find the matrix of the transformation obtained by carrying out T1 followed by T2.

a) The transformation T1 represented by the matrix

is an anticlockwise rotation about the origin. Find the angle of rotation.

b) The transformation T2 represented by the matrix

is a reflection in the line y = mx. Find the value of m.

c) (i) The matrix M3 is given by M3 = M1M2. Find M3. (ii) The matrix M3 represents a single transformation T3. Give a geometrical description of T3.

The transformation S is represented by the matrix and the transformation T is represented by the matrix .
(i) Give geometrical descriptions of the transformations S and T.
(ii) Find the single matrix, C, which represents transformation S followed by transformation T.
(iii) Hence find matrices X and Y such that C-1 = XY, where neither X nor Y is a multiple of the unit matrix.

Interpreting the determinant of the transformation matrix

Suppose that a transformation T is represented by the matrix M.
When a shape is transformed by this transformation matrix, the area of the image is related to the area of the object according to this relationship:

.

Therefore |det(M)| represents the area scale factor for the transformation.

Note: If det(M) is negative, then the transformation will have involved some flipping over (i.e. some reflection is involved).

Worked examination question (OCR)
The matrix A is given by .
a) Describe fully a sequence of two geometrical transformations represented by A.
b) The triangle PQR is mapped by the transformation represented by A onto the triangle P’Q’R’. Given that the area of PQR is 8, find the area of P’Q’R’.

Solution
a) Consider the image of the unit square under the transformation, The image of the point (1, 0) is (1, √3), i..e the first column of the matrix
The image of the point (0, 1) is (-√3, 1), i.e. the second column of the matrix.
The image of the point (1, 1) is (1-√3, √3+1), i.e. the sum of the entries

We can see that two transformations are involved: The unit square has been enlarged; The unit square has been rotated.

The length of the sides of the enlarged square are
So the scale factor of the enlargement must be 2.

The angle of rotation is the angle θ shown on the diagram above.
Using trigonometry we find that θ = 60°.

So the two transformations are: * enlargement s.f. 2 centre (0, 0) * rotation 60° (anticlockwise) centre (0, 0).

b) To find the area of the image, we need to find det(A). This determinant is 1 – (-3) = 4.
So the area of the image is 4 times the area of the object.
Therefore the area of P’Q’R’ is 4 × 8 = 32.

Mixed questions
a) Find the 2 × 2 matrix which represents a clockwise rotation through an angle of θ about the origin.
b) Find the matrix which transforms
.

Solution:
a) The matrix (as given in the formula book) represents a rotation through an angle of θ in an anticlockwise direction.

A rotation of θ in a clockwise direction is the same as a rotation of –θ in an anticlockwise direction.

Therefore the required matrix is .

b) We wish to find a matrix M such that
.
This can be written as a single equation:
.

Therefore .

But .

So, =0
Find the matrix which transforms
.