Grade 9 IGCSE A1: Chapter 9 Matrices and Transformations Materials Needed: Straightedge, Graph Paper Exercise 1: Matrix Operations Matrices are used in Linear Algebra to solve systems of linear equations. The next simple example takes a look at a system of equations like and changes its form to 3x y = 8 3 1 x + 3y = 6 3 x y = 8 6 Solution Methods: a. Elimination b. Substitution c. Graphical Solution Method: a. Matrix Method Therefore, after becoming familiar with the Matrix Method, you will be able to solve a system of two linear equations in four different ways. So let s understand what Matrices are and how to perform operations on them. Definition: A matrix is a rectangular array of numbers arranged in rows and columns. Plural: Matrices Symbolized by a Capital bold letter, example A (you will not need to write in bold, but be aware of the notational difference What does a non-bold upper case letter usually symbolize?) An example of a X (two by two) matrix: A = 1 or A = 1 3 4 3 4 Each matrix has a dimension/order: (number of rows) by (number of columns). What is the dimension/order of the following matrix? (1 3) A matrix with the same number of rows and columns is called a? matrix. Each number in the array is called an element of the matrix. Each element (symbolized by an italicized lower case letter) is designated by its position in the matrix ie. Row x, column y or say a3 means the nd row, 3 rd column element. A column matrix (has only one column), a row matrix (has only one row) and zero matrix has only zeros as its elements. Matrix equality: Two matrices are equal iff they have the same order and corresponding elements. Example: 1 = 1 but 1 1 3 4 3 4 3 4 3 4 How might you express matrix equality using matrix notation? A = B Matrix addition and subtraction: Add/subtract corresponding elements of only same order matrices. Example: 1 4 + 1 = 3 4 3 4 6 8 Example: 1 + (1 3) cannot be added together as the dimensions/orders are not the same. 3 4 Example: 1 0 1 = 0 = O 3 4 3 4 0 0 In general: A aaa ±B aaa = P aaa Notice that A + (-A) = O. Since adding A and its opposite A produces the zero matrix which we called the additive identity element, A and A are called inverse elements of each other because of the following fact: A + (-A) = O AND (-A) + A = O. This should remind you of the inverse property of functions.
Scalar multiplication: Multiply all elements by the scalar. There is no such thing as scalar division. Example: 1 4 4 = 1 + 1 = therefore 1 = 3 4 3 4 3 4 6 8 3 4 6 8 In other words, if A = 1 4, then A = 3 4 6 8 Do not write: 4 6 8 as this notation does not exist. How should you write it? Matrix multiplication: Take a look at the textbook page 9 regarding x matrix multiplication. The orders of the two matrices must satisfy the following property, otherwise matrix multiplication is not possible: A mmm B ppp Also, if matrix multiplication is possible, then the order of the resultant matrix will be A mmm B ppp = R mmm. Let s practice some Matrix Multiplication Notice that positive integer exponents to represent repeated matrix multiplication still holds true: AA = A Is matrix multiplication commutative? In other words, is AA = BB always true? If you can find one example when it is not true, then in mathematics, you cannot say that it is true (even though it may be true sometimes or even most of the times). As a result, if we multiply matrix A to B, we have to state whether we are multiplying A to the left or right of B as the resultant matrix is different. Left multiply A to B AB Right multiply A to B BA There is no such thing as Matrix Division 1 3 4 Exercise 1: C-type: 3 36 last column; 38 46 last column A-type: 48, 49 (1 3) is meaningless.
Exercise : Inverse of a Matrix The notation for the inverse of a matrix is similar to that of a function. The inverse matrix of A is written as A 1. What we want to do now is demonstrate the matrix method to solve systems of linear equations. 3x y = 8 x + 3y = 6 Solution Solution Method: Remember though that we said that we Methods: a. Matrix Method cannot divide by a matrix. Therefore we a. Elimination have to introduce a matrix that is called the b. Substitution Identity Matrix represented by I that is a c. Graphical square matrix with only two different elements, namely 0 and 1. All the 1 s are located on the diagonal that goes from the top left to the bottom right. All the other elements are zero. Examples: I = 1 0 1 0 0 0 1 and I 333 = 0 1 0. For the IGCSE course, we will only be using I = 1 0 0 1. 0 0 1 The important multiplicative property of I is that if you multiply it to any matrix B, it will not change the identity of B. Prove to yourself that this is true. We need to investigate the inverse property. Recall the inverse property of functions and inverse functions: f f 1 (x) = x = f 1 (f(x)) Given that the multiplicative inverse of A, if it exists, it is written as A 1 and satisfies AA 1 = I = A 1 A. So in order to solve 3 1 3 x y = 8 we will write this in this form: AA = B and solve for X. 6 AA = B In order to solve for X, as there is no matrix division, we will left multiply each side by A 1. Notice what left multiply means in the next line. A 1 AA = A 1 B II = A 1 B X = A 1 B So the question becomes how to solve for A 1 if you are given an A. 3 1 3 x y = 8 6 If A = a b c d, then A 1 = 1 d b aa bb c a. We can also write it as A 1 = 1 d b A c called the determinant of A also written as deta. where A = aa bb is a If A is not equal to zero, there is a unique solution to the equation AX=B. In this case, we say that the matrix A is non-singular and A has an inverse matrix. If A is equal to zero, there is no unique solution to the equation AX=B. In this case, we say that the matrix A is singular and A does not have an inverse matrix. Given two linear equations, when does the system not have a solution or an infinite number of solutions? Provide two systems that represent this. What is evident from the matrix form of the solution? Exercise : C-type: 3, 9, 15, 17, 19, 1,, 5 A-type: 4, 6, 8
For all the Simple Transformations, use the links to see more examples and do activities on the web to ensure that your understanding of the concepts is correct. Exercise 3-4: Simple Transformations (Reflections) A Graphical Transformation is a mathematical means by which to change a given object s position, orientation or shape in a consistent way. Transformations are categorized as either Rigid or Non-Rigid Transformations. Name of Transformation Reflection Reflection in D Wolfram Demo Needed Graphically Algebraic Line of Reflection Must be able to draw lines of reflections based upon equations. Primary Reflection Lines: Horizontal lines: y = a Vertical Lines: x = b Diagonal Lines: y = ±x + c The line of reflection is denoted with a dotted line and labeled. The reflection is always perpendicular to the mirror line and on the opposite side of the mirror. 1 Triangle 1 has been transformed to Triangle. Coordinates are labelled in this way: A(a,b) A (c,d) A is mapped onto A Notice the similar language with the functional notation f: x x + 1 Exercise 3-4: C-type: Exercise 3 3, 6, 9 Exercise 4 3 A-type: Exercise 4 4
Exercise 5-6: Simple Transformations (Rotations) Name of Transformation Rotation Rotations Wolfram Demo Needed Graphically Algebraic Center of Rotation Angle of Rotation Direction of Rotation Clockwise Negative angle measure Counterclockwise Positive angle measure Label Center as O 1 3 Triangle 1 has been transformed to Triangle which in turn has been transformed to Triangle 3. Coordinates are labelled in this way; A(a,b) A (c,d) A (e,f) A is mapped onto A which is mapped onto A In order to find the center of Rotation given an original image and a transformed image via rotation, if you construct the perpendicular bisectors connecting corresponding coordinates, the intersection will be the center of rotation. How would you determine the angle of rotation? Exercise 5-6: C-type: Exercise 5, 4, 7 Exercise 6 3 A-type: Exercise 6 5
Exercise 7-9: Simple Transformations (Translation and Enlargement and all Transformations Combined) Name of Transformation Translation Needed Graphically Algebraic Horizontal Translation Vertical Translation Combined into a Column Vector A translation is a slide of the image without changing the orientation or dimensions of the image. x y Enlargement Enlargements Wolfram Demo Center of Enlargement Scale Factor = k Center of Enlargement designated with coordinate O. Dotted lines emanating from O to the respective vertices of the shape. Investigate different values of k: If k > 1, then If k = 1, then If 0 < k < 1, then If 1 < k < 0, then If k < 1, then Investigate different values of k: What happens to the image? What happens to the image? What happens to the image? What happens to the image? What happens to the image? In order to find center of an enlargement, connect corresponding coordinates and the intersection of the lines will be the center of enlargement. How would you determine the scale factor? Exercise 7-8: C-type: Exercise 7 1 Exercise 8, 5 Exercise 9 1 A-type: Exercise 8 11, 13
Exercise 10-11: Combined and Inverse Transformations Now that we ve gone through all the Simple Transformations, how would you categorize each transformation Rigid Non-Rigid Combined Transformations Use Matrices (and simple column vector) to represent Transformations: T Translation R Rotation F Reflection E Enlargement Repeated Transformations Inverse Transformations Will undo the original transformation. Needed Graphically Algebraic A shape to transform (usually a triangle) All information required to perform each transformation All pertinent notation on the graph. As there will be combined transformations, the order in which you perform them is key. Coordinates are labelled in this way; A(a,b) A (c,d) A (e,f) A is mapped onto A which is mapped onto A Inverse Transformation Notation: T 1 Translation R 1 Rotation F 1 Reflection E 1 Enlargement We used this: 1 3 Now, we will designate what transformation is being applied to the shape. Now: T( 1) = So let s say we do this: RR( 1) = R( ) = 3 What does the statement mean? R ( 1) = RR( 1) Recall that the exponents can only be positive integer values. If T = a b, then T 1 = If R is given with Center at O and 90 degree rotation, what is R 1? If F is given with line of reflection y = a, what is F 1? If E is given with Center at O and SF = k, what is E 1? Notation: T 3 ( 1) = (T 1 ) 3 ( 1) = T 1 T 1 T 1 ( 1) Exercise 10-11: C-type: Exercise 10 4 0 last column A-type: Exercise 11 9, 10
Exercise 1, then 13: Transformation Matrices (Two Days) Now let s use Matrices to mathematically determine where the transformed coordinates will be located. Graphical Interpretation/Description Reflect the point A(3, 5) and B(-1, -) about the line y = 0. The transformed point is A (3, -5) and B (-1,) The Transformation F can be written in the following matrix form: F = 1 0 0 1 Transformation Matrices applied to a shape Wolfram Demo Algebraic Interpretation IS: T x 1 x = x 1 x y 1 y T represents either R Rotation F Reflection E Enlargement x 1 x represents two coordinate values of the original shape. x 1 x represents the corresponding two coordinate values of the y 1 y transformed shape. T x 1 x represents the matrix multiplication of the two matrices. Using the example to the left: IS: T x 1 x = x 1 x y 1 y 1 0 1 3 1 3 = 0 1 5 5 Therefore the Transformed point is A (3, -5) Translation Matrix Translations Matrices do not involve matrix multiplication as they merely translate using a vector. Translate the point A(3, 5) to the right 1 unit and down 4 units. The transformed point is A (4, 1) The Transformation T is written in the following vector form: T = 1 4 IS: x 1 y 1 + T = x 1 y 1 T represents the Translation Vector x 1 y 1 represents a coordinate value of the original shape. x 1 y 1 represents the corresponding coordinate value of the transformed shape. x 1 y 1 + T represents the addition of the two matrices Using the example to the left: IS: x 1 y + T = x 1 1 y 1 3 5 + 1 4 = 4 1 Therefore the Transformed point is A (4, 1)
While you are doing these exercises, try to determine the link between the Matrix form of a Translation and the Translation itself. Exercise 1: C-type: Exercise 1, 5 A-type: Exercise 1 9, 11cg Next Day Exercise 1-13: C-type: Exercise 13, 4a, 5 A-type: Exercise 13 6, 7
Exercise 14: Determining the Matrix form of a transformation Use base vectors to determine the transformation matrix T x. Description of Transformation Graphical Interpretation Algebraic Interpretation Transformation that does not change anything. Identity Matrix Start with the Identity Matrix form, then geometrically apply the transformation to the basis vectors. Then place the colored basis vectors in their respective location in the matrix. I = 1 0 0 1 R 90 aaaaa (0,0) = 0 1 1 0 R 90 aaaaa (0,0) Rotation of 90 degrees about the Origin (0, 0) Follow the directions as the rotation but this time apply a reflection. F y=x Reflection about the line y = x Follow the directions as the rotation but this time apply an enlargement. E SS= aaaaa (0,0) Enlargement for Scale Factor - about the Origin (0, 0) F y=x = 0 1 1 0 E SS= aaaaa (0,0) = 0 0 = 1 0 0 1 = I Notice: If your transformations do not include the Origin, then the matrix forms of the transformations tend to be more complicated. In order to still apply matrix transformations, it is matter of translating to center at (0, 0), apply the desired transformations and then apply an inverse translation vector. Also, if you know the original and transformed coordinates of the IS: T x 1 x = x 1 x, then you can apply y 1 y a right multiply by x 1 x 1 to both sides of the equation to solve for T to determine the transformation matrix. In other words, T x 1 x x 1 x 1 = x 1 x y 1 y TT = x 1 x x 1 x y 1 y 1 T = x 1 x x 1 x y 1 y 1 x 1 x 1 Exercise 14: C-type: 1 18 all A-type: Find the Transformation Matrix equation that will perform the following transformation and then apply your transformation matrix equation on A(4,5) to find the coordinate of A.
a. A rotation of 90 degrees about O(1, ) b. An enlargement with SF = - about O(1, ) b. A reflection about y = x + 3