Matrix Transformations

Matrix Transformations

Introduction
Having been introduced to transformations and matrices earlier on, you will now represent these transformations in matrix form. Matrices can be viewed as linear mappings. The transformations of geometric shapes can be achieved using matrix multiplication, which is an important concept in computer animation, robotics and calculus. This concept of matrix transformation is also used in computer science
while designing the user’s interface. With the knowledge acquired in this topic, you will be able to transform shapes on a coordinate grid using matrices. 9.1 Identifying Transformation Matrices for Reflection.
Rotation and Enlargement In Senior Two, you discussed the various types of transformations, which included translations, reflections, enlargements and rotations. In this subtopic, you will discover
the connection between matrices and transformations.

Identifying transformation matrices for reflection
Activity 9.1 (a) (Work in groups)
(a) Reflect the points P(2, 1) and Q(-3, 5) on a Cartesian plane along the x-axis
and the y-axis.
(b) Write down the image points after each reflection.
(c) What do you realise about objects reflected along those axes?
Activity 9.1 (b) (Work in groups)
(a) Plot the points A(O, 0), B(l, 0), C(l, 1) and D(O, 1) on a graph paper.
(b) Record the coordinates of points B and D as a 2 x 2 matrix. What do you notice?
(c) Join the points plotted in (a) to form a square.
(d) Reflect your square along the x-axis.
(e) Record the coordinates of the images of points B and D as a 2 x 2 matrix.
(f) What do you conclude about the matrix formed in (e) and its determinant?
Activity 9.1 (c) (Work in groups)
(a) Reflect the square drawn in Activity 9. 7(b) part(c) along the y-axis.
(b) Record the coordinates of the images of points B and D as a 2 x 2 matrix.
(c) What do you conclude about the matrix formed in (b) and its determinant?
Activity 9.1 (d) (Work in groups)

(a) Determine the matrix of reflection by considering the images of the points
1(1, 0) and J(O, 1) after a reflection along the line;
(b) What do you notice about the determinants of the transformation matrices?

Activity 9.1 (f) (Work in groups)
(a) Draw the image triangle of triangle ABC with vertices A(l, 2), B(5, 3), C(4, 6)
under a rotation of 900 about the origin in the;
i) anti-clockwise direction
ii) clockwise direction
g (b) Record the image of each point in each of the rotations.
(c) Find the determinants of the matrices of transformation.
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(d) Study and write your observations about the determinants obtained in (c) above.
Activity 9.1 (g) (Work in groups)
; (a) Find the images of rotation of 1(1, 0) and J(O, 1) about the origin in the
positive half turn.
(b) Hence, determine the matrix of transformation.
(c) Present your findings to the rest of the class.
Identifying transformation matrices for enlargement
Activity 9.1 (h) (Work in groups)

Activity 9.1 (i) (Work in groups)
(a) Find the images of the points 1(1, 0) and J(O, 1) when enlarged with scale
factor 4 about the origin. Hence, state the transformation matrix.
(b) What do you notice about the determinant of the transformation matrix?
(c) What if the scale factor was k? What would be the transformation matrix?

9.4 Determining the Inverse of a Transformation Matrix
In the previous topic, you learnt how to determine the inverse of a matrix. Now, you
will use the same ideas to determine the inverse of a transformation matrix.

Activity 9.4 (Work in groups)

9.6 Identifying the relationship between Area Scale Factor
and Determinant of Transformation Matrix Under this subtopic, you will discover the relationship between area scale factor and determinant of a transformation matrix.

9.7 Determining a Single Matrix for Successive Transformations When a linear transformation Tl is followed by a second linear transformation T 2, the two transformations can be represented by a single matrix T.

ICT Activity (In groups)
(a) Type a report of not more than 300 words about the application of matrix transformation in real life. Remember to include illustrations, where necessary. (b) Make a printout of your work.