Wednesday, January 25, 2023

MATHEMATICS FORM FOUR TOPIC 7: MATRICES AND TRANSFORMATIONS

MATHEMATICS FORM FOUR TOPIC 7: MATRICES AND TRANSFORMATIONS

MATHEMATICS FORM FOUR TOPIC 7: MATRICES AND TRANSFORMATIONS
Wednesday, January 25, 2023

The transformation matrix transforms a vector into another vector, which can be understood geometrically in a two-dimensional or a three-dimensional space. The frequently used transformations are stretching, squeezing, rotation, reflection, and orthogonal projection.

MATHEMATICS FORM FOUR TOPIC 7: MATRICES AND TRANSFORMATIONS


MATHEMATICS FORM FOUR TOPIC 7: MATRICES AND TRANSFORMATIONS

Introduction
  • 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 1 
    matrix1 example 1
    If the vector  matrix 2 example 1 
    is thought of as a position 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 matrix example 1Then T maps points (x, y) onto image points x,y
    matrix4 example 1  

Finding the Matrix of Transformation

  • The objective is to find the matrix of given transformation.

    Example 2
    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 triangles is P1(1,-3,Q1(3,-3and R1(2,-5).

    Solution
    Let the matrix of the transformation be matrix abcd

    Equating the corresponding elements and solving simultaneously
    matrix 2 example 2
    a + 3b = 1
    3a + 3b = 3
    2a= 2
    a = 1 and b = 0
    c + 3d = -3
    3c + 3d = -3
    2c= 0
    c = 0 and d = -1
    Therefore the transformation matrix is Transformation matrix example 2

    Example 3
    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 A1(-4,1,B1(-1 ,3,C1(-1,5,D1(-4 ,7).Describe the transformation and find its matrix

    Solution
    Let the matrix of the transformation be matrix abcd
    Equating the corresponding elements we get;
    matrix 1 example 3
    a + 4b = - 4                   c + 4d = 1
    3a + b = -1                    3c + d = 3

    Solve the equations simulteneously
    3a + 12b = - 12
    3a + b = - 1
    11b = -11
    hence b =-1 or a = 0

    3c + 12d = 3
    3c + d =3
    11d = 0
    d = 0 c = 1
    The matrix of the transformation is therefore Transformation matrix example 3
    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
unit square

  • The unit square ABCD with vertices A (0,0,B(1,0,C(1,1and D(0,1helps us to get the transformation of a given matrix and also to identify what trasformation a given matrix represent.

    Example 4
    Find the images of I and J under the trasformation whose matrix is;
    example unit square
    Solution
    solution matrix unit square
    solution 2 matrix unit square

    NOTE;
  • The images of I and J under transformation represented by any 2 x 2 matrix i.e., matrix abcdare I1(a ,c) and J1(b ,d)

    Example 5
    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 Transformation matrix example 2
  • unit square

    Example 6
    Show on a diagram the unit square and it image under the transformation represented by the matrix matrix 1 example 6
    diagram example 6

    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 matrix k example 6
    Therefore ,K
    1,the image of K is ( 5 ,1)

 

Successive Transformations

  • The process of performing two or more transformations in order is called successive transformation e.g. 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
    successive tranformations matrix 1
  • This transformation matrix creates a reflection in the x-axis. When multiplying by this matrix, the x coordinate remains unchanged, but the y co-ordinate changes sign.
    successive tranformations matrix 2
  • This transformation matrix creates a reflection in the y-axis. When multiplying by this matrix, the y coordinate remains unchanged, but the x co-ordinate changes sign.
    successive tranformations matrix 3
  • This transformation matrix creates a rotation of 1 80 degrees. When multiplying by this matrix, the point matrix is rotated 1 80 degrees around (0, 0). This changes the sign of both the x and y co-ordinates.
    successive tranformations matrix 4
  • This transformation matrix creates a reflection in the line y=x. When multiplying by this matrix, the x coordinate becomes the y co-ordinate and the y-ordinate becomes the x co-ordinate
    successive tranformations matrix 5
  • 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).
    successive tranformations matrix 6
  • 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).
    successive tranformations matrix 7
  • 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.
    successive tranformations matrix 8

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 T1,then a transformation that can map T1 back to T is called the inverse of the transformation A , written as image A-1.
  •  If R is a positive quarter turn about the origin the matrix for R is inverse matrix 1 and the matrix for R-1 is inverse matrix 2 hence R-1R = RR-1 = 1 

    Example 7
    T is a triangle with vertices A (2, 4), B (1 , 2) and C (4, 2).S is a transformation represented by the matrix matrix 1 example 7
    1. Draw T and its image T1 under the transformation S
    2. Find the matrix of the inverse of the transformation S

    Solution
    1. Using transformation matrix S =matrix 1 example 7
      answer example 7
    2.  Let the inverse of the transformation matrix be(ac db). This can be done in the following ways
      1. S-1S = 1
        Therefore
         matrix 1 example 7b

        Equating corresponding elements and solving simultaneously;
        a = 1 ,b = -2 , c = 0 and d = 2
        Therefore
        matrix 2 example 7b

         matrix 3 example 7b
      2.  
        matrix example 7bII

Area Scale Factor and Determinant of Matrix

  • The ratio of area of image to area object is the area scale factor (A.S.F)
    Area scale factor = area of image
                                area of object
  • Area scale factor is numerically equal to the determinant. If the determinant is negative you simply ignore the negative sign.

    Example 8
    Area of the object is 4 cm and that of image is 36 cm find the area scale factor.

    Solution

    36 = 9
    4
    If it has a matrix of  matrix example 8the determinant is 9 - 0 = 9 hence equal to A.S.F

Shear and Stretch

Shear

  • The transformation that maps an object (in orange) to its image (in blue) is called a shear
    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;
    1. The invariant line
    2. A point not on the invariant line, and its image.

    Example 9
    A shear X axis invariant
    shear example 1
    Example 10
    A shear Y axis invariant
    Shear example 2i
    shear example 2ii

    Note;
  • Shear with x axis invariant is represented by a matrix of the form shear matrix form x invariant under this trasnsformation, J (0, 1 ) is mapped onto J1(k,1).
  • Likewise a shear with y – axis invariant is represented by a matrix of the form shear matrix form y invariant. Under this transformation, I (0,1 ) is mapped onto I1(1,k).

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;
    1. The scale factor
    2. The invariant line

  • Note;
    1. If K is greater than 1 , then this really is a stretch.
    2. If k is less than one 1 , it is a squish but we still call it a stretch
    3. If k = 1 , then this transformation is really the identity i.e. it has no effect.

    Example 11
    Using a unit square, find the matrix of the stretch with y axis invariant ad scale factor 3
    stretch example
    Solution
    The image of I is I1(1, 0and the image of J is (0,1) therefore the matrix of the stretch is stretch matrix answer
  • Note;
    The matrix of the stretch with the y-axis invariant and scale factor k is stretch matrix form y axis invariantand the matrix of a stretch with x – axis invariant and scale factor k is stretch matrix form x axis invariant

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

Past KCSE Questions on the Topic

  1. Matrix p is given by matrix p kcse questionsFind P-1
  2. A triangle is formed by the coordinates A (2, 1 ) B (4, 1 ) and C (1 , 6). It is rotated clockwise through 90about the origin. Find the coordinates of this image.
  3. On the grid provided, A (1 , 2) B (7, 2) C (4, 4) D (3, 4) is a trapezium
    grid kcse quetion 3
    1. ABCD is mapped onto A’B’C’D’ by a positive quarter turn. Draw the image A’B’C’D on the grid
    2. A transformation matrix tranformation matrix question 3bmaps A’B’C’D onto A”B” C”D” Find the coordinates 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 Twhose vertices are A(-4, 3) B(-1 , 3) and C(x, y) by a Transformation M = matrix abcd
    1. Find the:
      1. Matrix M of the transformation
      2. Coordinates of C1
    2. Triangle Tis the image of triangle Tunder a reflection in the line y = x. Find a single matrix that maps T and T2
  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=matrix abcdFollowed by matrix p question 5
    1. Find R
    2. 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 coordinate’s plane below
    triangle abc q6
    1. Given that A (-6, 5) is mapped onto A (6,-4) by a shear with y- axis invariant
      1. Draw triangle A’B’C’, the image of triangle ABC under the shear
      2. Determine the matrix representing this shear
    2. Triangle A B C is mapped on to A” B” C” by a transformation defined by the matrix matrix q6b
      1. Draw triangle A” B” C”
      2. Describe fully a single transformation that maps ABC onto A”B” C”
  7. Determine the inverse T-1of the matrix T = matrix T q7
    Hence find the coordinates to the point at which the two lines x + 2y = 7 and x - y =1
  8. Given that A = matrix A q8B =matrix B q8
    Find the value of x if
    1. A- 2x = 2B
    2. 3x – 2A = 3B
    3. 2A - 3B = 2x
  9. The transformation R given by the matrix
    matrix q9

    1. Determine the matrix A giving a, b, c and d as fractions
    2. Given that A represents a rotation through the origin, determine the angle of rotation. 
    3. S is a rotation through 180 about the point (2, 3). Determine the image of (1 , 0) under S followed by R
MATHEMATICS FORM FOUR TOPIC 7: MATRICES AND TRANSFORMATIONS
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