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Specific Objectives

By the end of the topic the learner should be able to:

(a) Relate image and object under a given transformation on the Cartesian

Plane;

(b) Determine the matrix of a transformation;

(c) Perform successive transformations;

(d) Determine and identify a single matrix for successive transformation;

(e) Relate identity matrix and transformation;

(f) Determine the inverse of a transformation;

(g) Establish and use the relationship between area scale factor and determinant of a matrix;

(h) Determine shear and stretch transformations;

(i) Define and distinguish isometric and non-isometric transformation;

(j) Apply transformation to real life situations.

Content

(a) Transformation on the Cartesian plane

(b) Identification of transformation matrix

(c) Successive transformations

(d) Single matrix of transformation for successive transformations

(e) Identity matrix and transformation

(f) Inverse of a transformations

(g) Area scale factor and determinant of a matrix

(h) Shear and stretch (include their matrices)

(i) Isometric and non-isometric transformations

(j) Application of transformation to real life situations.

Matrices of transformation

A transformation change the shape, position or size of an object as discussed in book two.

Pre –multiplication of any 2 x 1 column vector by a 2 x 2 matrix results in a 2 x 1 column vector

Example

If the vector is thought of as apposition vector that is to mean that it is representing the points with coordinates (7, -1) to the point (17, -9).

Note;

The transformation matrix has an effect on each point of the plan. Let’s make T a transformation matrix T Then T maps points (x, y) onto image points

T

Finding the Matrix of transformation

The objective is to find the matrix of given transformation.

Examples

Find the matrix of transformation of triangle PQR with vertices P (1, 3) Q (3, 3) and R (2, 5).The vertices of the image of the triangle sis.

Solution

Let the matrix of the transformation be

=

Equating the corresponding elements and solving simultaneously

2a= 2

2c= 0

Therefore the transformation matrix is

Example

A trapezium with vertices A (1 ,4) B(3,1) C (5,1) and D(7,4) is mapped onto a trapezium whose vertices are .Describe the transformation and find its matrix

Solution

Let the matrix of the transformation be

Equating the corresponding elements we get;

Solve the equations simulteneously

11b = -11 hence b =-1 or a = 0

3c + d =3

The matrix of the transformation is therefore

The transformation is positive quarter turn about the origin

Note;

Under any transformation represented by a 2 x 2 matrix, the origin is invariant, meaning it does not change its position.Therefore if the transformtion is a rotation it must be about the origin or if the transformation is reflection it must be on a mirror line which passses through the origin.

The unit square

The unit square ABCD with vertices A helps us to get the transformation of a given matrix and also to identify what trasformation a given matrix represent.

Example

Find the images of I and J under the trasformation whose matrix is;

Solution

NOTE;

The images of I and J under transformation represented by any 2 x 2 matrix i.e., are

Example

Find the matrix of reflection in the line y = 0 or x axis.

Solution

Using a unit square the image of B is ( 1, 0) and D is (0 , -1 ) .Therefore , the matrix of the transformation is

Example

Show on a diagram the unit square and it image under the transformation represented by the matrix

Solution

Using a unit square, the image of I is ( 1 ,0 ), the image of J is ( 4 , 1),the image of O is ( 0,0) and that of K is

Successive transformations

The process of performing two or more transformations in order is called successive transformation eg performing transformation H followed by transformation Y is written as follows YH or if A , b and C are transformations then ABC means perform C first ,then B and finally A , in that order.

The matrices listed below all perform different rotations/reflections:

This transformation matrix is the identity matrix. When multiplying by this matrix, the point matrix is unaffected and the new matrix is exactly the same as the point matrix.

This transformation matrix creates a reflection in the x-axis. When multiplying by this matrix, the x co-ordinate remains unchanged, but the y co-ordinate changes sign.

This transformation matrix creates a reflection in the y-axis. When multiplying by this matrix, the y co-ordinate remains unchanged, but the x co-ordinate changes sign.

This transformation matrix creates a rotation of 180 degrees. When multiplying by this matrix, the point matrix is rotated 180 degrees around (0, 0). This changes the sign of both the x and y co-ordinates.

This transformation matrix creates a reflection in the line y=x. When multiplying by this matrix, the x co-ordinate becomes the y co-ordinate and the y-ordinate becomes the x co-ordinate.

This transformation matrix rotates the point matrix 90 degrees clockwise. When multiplying by this matrix, the point matrix is rotated 90 degrees clockwise around (0, 0).

This transformation matrix rotates the point matrix 90 degrees anti-clockwise. When multiplying by this matrix, the point matrix is rotated 90 degrees anti-clockwise around (0, 0).

This transformation matrix creates a reflection in the line y=-x. When multiplying by this matrix, the point matrix is reflected in the line y=-x changing the signs of both co-ordinates and swapping their values.

Inverse matrix transformation

A transformation matrix that maps an image back to the object is called an inverse of matrix.

Note;

If A is a transformation which maps an object T onto an image ,then a transformation that can map back to T is called the inverse of the transformation A , written as image .

If R is a positive quarter turn about the origin the matrix for R is and the matrix for is hence

Example

T is a triangle with vertices A (2, 4), B (1, 2) and C (4, 2).S is a transformation represented by the matrix

- Draw T and its image under the transformation S
- Find the matrix of the inverse of the transformation S

Solution

- Using transformation matrix S =

- Let the inverse of the transformation matrix be. This can be done in the following ways
Therefore

Equating corresponding elements and solving simultaneously;

Therefore

Area Scale Factor and Determinant of Matrix

The ratio of area of image to area object is the area scale factor (A.S.F)

Are scale factor =

Area scale factor is numerically equal to the determinant. If the determinant is negative you simply ignore the negative sign.

Example

Area of the object is 4 cm and that of image is 36 cm find the area scale factor.

Solution

If it has a matrix of

Shear and stretch

Shear

The transformation that maps an object (in orange) to its image (in blue) is called a shear

The object has same base and equal heights. Therefore, their areas are equal. Under any shear, area is always invariant ( fixed)

A shear is fully described by giving;

- The invariant line
- A point not on the invariant line, and its image.

Example

A shear X axis invariant

Example

A shear Y axis invariant

Note;

Shear with x axis invariant is represented by a matrix of the form under this trasnsformation ,J (0, 1) is mapped onto .

Likewise a shear with y – axis invariant is represented by a matrix of the form( ) . under this transformation, ,I (0,1) is mapped onto .

Stretch

A stretch is a transformation which enlarges all distance in a particular direction by a constant factor. A stretch is described fully by giving;

- The scale factor
- The invariant line

Note;

i.)If K is greater than 1, then this really is a stretch.

ii.) If k is less than one 1, it is a squish but we still call it a stretch

iii.)If k = 1, then this transformation is really the identity i.e. it has no effect.

Example

Using a unit square, find the matrix of the stretch with y axis invariant ad scale factor 3

Solution

The image of I is therefore the matrix of the stretch is

Note;

The matrix of the stretch with the y-axis invariant and scale factor k is and the matrix of a stretch with x – axis invariant and scale factor k is

Isometric and Non- Isometric Transformation

Isometric transformations are those in which the object and the image have the same shape and size (congruent) e.g. rotation, reflection and translation

Non- isometric transformations are those in which the object and the image are not congruent e.g., shear stretch and enlargement

End of topic

Did you understand everything? If not ask a teacher, friends or anybody and make sure you understand before going to sleep! |

Past KCSE Questions on the topic.

1. Matrix p is given by 1 2

4 3

(a) Find P^{-1}

(b) Two institutions, Elimu and Somo, purchase beans at Kshs. B per bag and

maize at Kshs m per bag. Elimu purchased 8 bags of beans and 14 bags of maize for Kshs 47,600. Somo purchased 10 bags of beans and 16 of maize for Kshs. 57,400

(c) The price of beans later went up by 5% and that of maize remained constant. Elimu bought the same quantity of beans but spent the same total amount of money as before on the two items. State the new ratio of beans to maize.

2. A triangle is formed by the coordinates A (2, 1) B (4, 1) and C (1, 6). It is rotated

clockwise through 90^{0} about the origin. Find the coordinates of this image.

3. On the grid provided on the opposite page A (1, 2) B (7, 2) C (4, 4) D (3, 4) is a trapezium

(a) ABCD is mapped onto A’B’C’D’ by a positive quarter turn. Draw the image A’B’C’D on the grid

(b) A transformation -2 -1 maps A’B’C’D onto A”B” C”D” Find the coordinates

0 1 of A”B”C”D”

4. A triangle T whose vertices are A (2, 3) B (5, 3) and C (4, 1) is mapped onto triangle T^{1} whose vertices are A^{1} (-4, 3) B^{1} (-1, 3) and C^{1} (x, y) by a

Transformation M = a b

c d

a) Find the: (i) Matrix M of the transformation

(ii) Coordinates of C_{1}

_{ }b) Triangle T^{2 }is the image of triangle T^{1} under a reflection in the line y = x.

Find a single matrix that maps T and T_{2}

5. Triangles ABC is such that A is (2, 0), B (2, 4), C (4, 4) and A”B”C” is such that A” is (0, 2), B” (-4 – 10) and C “is (-4, -12) are drawn on the Cartesian plane

Triangle ABC is mapped onto A”B”C” by two successive transformations

R = a b

c d Followed by P = 0 -1

-1 0

(a) Find R

(b) Using the same scale and axes, draw triangles A’B’C’, the image of triangle ABC under transformation R

Describe fully, the transformation represented by matrix R

6. Triangle ABC is shown on the coordinates plane below

(a) Given that A (-6, 5) is mapped onto A (6,-4) by a shear with y- axis invariant

- Draw triangle A’B’C’, the image of triangle ABC under the shear
- Determine the matrix representing this shear

(b) Triangle A B C is mapped on to A” B” C” by a transformation defined by the matrix -1 0

1½ -1

(i) Draw triangle A” B” C”

(ii) Describe fully a single transformation that maps ABC onto A”B” C”

7. Determine the inverse T^{1} of the matrix 1 2

1 -1

Hence find the coordinates to the point at which the two lines

x + 2y = 7 and x – y =1

8. Given that A = 0 -1 and B = -1 0

3 2 2 -4

Find the value of x if

(i) A- 2x = 2B

(ii) 3x – 2A = 3B

(iii) 2A – 3B = 2x

9. The transformation R given by the matrix

A = a b maps 17 to 15 and 0 to -8

c d 0 8 17 15

(a) Determine the matrix A giving a, b, c and d as fractions

(b) Given that A represents a rotation through the origin determine the angle of rotation.

(c) S is a rotation through 180 about the point (2, 3). Determine the image of (1, 0) under S followed by R.