Whiteboardmaths.com © 2011 All rights reserved 5 7 2 1

Intro Matrix Transformations 8 A transformation can be represented by a matrix. y r 6 Consider the point p(3,4) in the plane. It can be written as the column matrix. p 4 P’ The point p, can be transformed by applying a matrix to it. For example. What effect is this particular matrix having on points in the plane? 2 x O -8 -6 -4 -2 2 4 6 8 It produces a reflection in the line y = x. - 2 Similarly with point q(-7,-5) - 4 q and with point r(-6,6) Intro - 6 r’ q’ - 8

Matrix Transformations 8 y Let us confirm this with the triangle shown. r 6 s t u We can write the co-ordinates of each vertex in a matrix like so: P 4 s t u P’ 2 Applying gives x O -8 -6 -4 -2 2 4 6 8 s’ t’ u’ s’ t’ u’ - 2 The matrix produces a reflection in the line y = x - 4 q - 6 r’ q’ - 8

In this particular case A = A-1 Matrix Transformations 8 y If a matrix A, transforms an object then the inverse matrix A-1 will map the image back onto the original object. r 6 s t u P 4 In our example if A = s’ t’ u’ P’ 2 x O -8 -6 -4 -2 2 4 6 8 In this particular case A = A-1 s’ t’ u’ - 2 s’ t’ u’ - 4 q The matrix gives a reflection in the line y = x - 6 r’ q’ - 8

c a b Matrix Transformations 8 y Question 1.  Reflection in y = x Matrix Transformations 8 y Question 1. Transform the rectangle shown by each of the matrices in turn. Draw and label the image in each case and describe the transformation. a b c a’ d’ b’ c’ c 6 a b 4 2 d c a x O -8 -6 -4 -2 2 4 6 8 Reflection in y = -x a’ b’ c’ d’ a d’ a’ b’ c’ b - 2 b 90o CW rotation about (0,0) - 4 c - 6 90o ACW rotation about (0,0) - 8

3 1 2 1  2 and 1 3 1  2 1  3 Matrix Transformations  Reflection in y = x Matrix Transformations 8 y Investigate which reflection matrices map: a a’ 6 1  2 and 1 3 Check your results for another shape of your choice. 3 4 1 b’ d d’ b 1  2 Reflection in x axis 2 c c’ x O c -8 -6 -4 -2 2 4 6 8 c’ - 2 d’ 2 b’ 1  3 Reflection in y axis - 4 - 6 a’ - 8

3 1 2 4 1  2 and 3 4 1  2 3  4 Matrix Transformations 8 y Investigate the rotation matrices that map: e a 6 1  2 and 3 4 f 3 Check your results for another shape of your choice. 4 1 d b 1  2 Rotation of 180o about (0,0) 2 h g c x O c -8 -6 -4 -2 2 4 6 8 c’ g’ h’ - 2 b’ 2 d’ 4 3  4 Rotation of 180o about (0,0) - 4 f’ - 6 a’ e’ - 8

a b Matrix Transformations 8 Question 2. y Transform the triangle shown by each of the matrices in turn. Draw and label the image in each case and describe the transformation. a b a’ b’ c’ a 6 4 a 2 b a c x O -8 -6 -4 -2 2 4 6 8 Enlargement by scale factor 2 from (0,0). a’ b’ c’ b - 2 b - 4 Enlargement by scale factor -2 from (0,0). In general the matrix produces an enlargement by scale factor k from (0,0) - 6 - 8

Enlargement by scale factor k from (0,0) Matrix Transformations RESULTS It is useful to try to commit these results to memory. x axis y axis y = x y = -x Reflections Rotations about (0,0) CW 90o ACW 90o CW or ACW 180o Enlargement by scale factor k from (0,0)

1 2 Matrix Transformations 8 y a The coordinates of any shape when multiplied by the identity matrix remain the same. e.g. In case you cannot remember all of these simple transformation matrices we now describe a method that helps you to derive them once you have determined what the transformation in question is. 6 4 1 d b 2 So if we apply a simple transformation to the identity matrix on a graph, we can describe the effect, and so find the transformation matrix. c x O -8 -6 -4 -2 2 4 6 c’ - 2 d’ 2 b’ To do this we view the identity matrix as two column vectors. - 4 - 6 a’ - 8

1 2 Matrix Transformations 8 y Consider the reflection in the x axis of kite 1 to kite 2. To obtain the transformation matrix we simply plot the base vectors of the identity matrix on the grid and view what happens to them under the transformation described. Base vectors a We simply recombine these two unit vectors (in order) to get the transformation matrix for a reflection in the x axis. 6 4 1 d b 2 c x O -8 -6 -4 -2 2 4 6 c’ After the reflection the vector remains in place - 2 d’ 2 b’ - 4 But  - 6 a’ - 8

1 2 Matrix Transformations 8 y a A 180o rotation about (0,0) 6  4 d  c x O Transformation matrix is -8 -6 -4 -2 2 4 6 c’ - 2 b’ 2 d’ - 4 - 6 a’ - 8

Matrix Transformations Use this method to write down the transformation matrices for: (a) Reflection in y = x (b) Enlargement from the origin by scale factor 3. (c) A 90o ACW rotation about (0,0) y 1 (a)  -1 x 1 (b)  (c)  -1

Matrix Transformations By visualising each transformation from our earlier results table when applied to the base vectors below, convince yourself that you can determine the matrix transformation in each case. (Could you work out all of these if the table was removed?) y 1 x -1 1 -1

Matrix Transformations AREA SCALE FACTORS and DETERMINANTS 8 y We need to establish the relationship between the two. a’ c’ b’ a’ c’ b’ Consider the triangle shown. We will transform it by each matrix in turn and compare resultant areas with the determinants. a b 6 Area = 4½ a 4 2 a Area = 9 c b x O -8 -6 -4 -2 2 4 6 8 image area = 4 x 4½ = 18 - 2 b The determinant of a transformation matrix gives the area scale factor between object and image. - 4 Area = 18 image area = 2 x 4½ = 9 - 6 - 8

Matrix Transformations AREA SCALE FACTORS and DETERMINANTS 8 y Confirm this statement by applying each matrix below to the square shown. Question 3 a b 6 Area = 16 a’ b’ d’ 4 a a b 2 Area of Square = 4 and 4 x 4 = 16 x d O c -8 -6 -4 -2 a’’ d’’ b’’ 2 4 6 8 b - 2 and 6 x 4 = 24 The determinant of a transformation matrix gives the area scale factor between object and image. - 4 - 6 Area = 24 - 8

Image area - 3 x 3 = -9  9 (ignoring negative sign) Matrix Transformations Inverse Matrices 8 y Worked Example: The matrix maps the triangle onto its image. (a) Draw the image. (b) Calculate the image area. (c) Calculate the matrix that maps the image back onto the object. 6 a’ b’ c’ 4 Area = 3 a b 2 c x (a) O -8 -6 -4 -2 2 4 6 8 (b) - 2 As discussed earlier, if a matrix A, transforms an object then the inverse matrix A-1 will map the image back onto the original object. Image area - 3 x 3 = -9  9 (ignoring negative sign) - 4 (c) - 6 - 8

Matrix Transformations Inverse Matrices 8 y Question 4 The matrix maps the parallelogram onto its image. (a) Draw the image. (b) Calculate the image area. (c) Calculate the matrix that maps the image back onto the object. a’ b’ c’ d’ 6 4 d a 2 Area = 6 x O c b -8 -6 -4 -2 2 4 6 8 (a) - 2 As discussed earlier, if a matrix A, transforms an object then the inverse matrix A-1 will map the image back onto the original object. (b) - 4 Image area 8 x 6 = 48 (c) - 6 - 8

Q1: Transform the rectangle shown by each of the matrices in turn Q1: Transform the rectangle shown by each of the matrices in turn. Draw and label the image in each case and describe the transformation. Q 2: Transform the triangle shown by each of the matrices in turn. Draw and label the image in each case and describe the transformation. a b c a b Q3: Confirm the statement below by applying each matrix below to the square shown. Q4: The matrix maps the parallelogram onto its image. (a) Draw the image. (b) Calculate the image area. (c) Calculate the matrix that maps the image back onto the object. “The determinant of a transformation matrix gives the area scale factor between object and image”. a b Worksheet