We need A to satisfy f ( x) = A . A transformation that slants the shape of an object is called the shear transformation. The elements of a matrix are specified by the row and column they reside in. Matrix Transformations - GeoGebra Matrix Transformations Author: Gareth Daniels Topic: Matrices, Reflection, Rotation Enter the transformation matrix by using the input boxes. This transformation matrix rotates the point matrix 90 degrees clockwise. If A is a transformation which maps an object T onto an image, 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, Find the matrix of the inverse of the transformation S. Equating corresponding elements and solving simultaneously; The ratio of area of image to area object is the area scale factor (A.S.F), Area scale factor is numerically equal to the determinant. Matrices can be used to represent linear transformations such as those that occur when two-dimensional or three-dimensional objects on a computer screen are moved, rotated, scaled (resized) or undergo some kind of deformation. You can add together two $2 \times 2$ matrices but not a $2 \times 3$ and a $2 \times 2$. So we can just add the third basis vector to our image without any transformation on it, . A transformation matrix that maps an image back to the object is called an inverse of matrix. 3. Astretch is described fully by giving; If K is greater than 1 , then this really is a stretch. In particular, check out the notes on matrix multiplication if you are unsure about how this linear transformation can be represented by the matrix; . Examples are rotations (about the origin) and reflections in some subspace. . a. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. To perform elementary transformations between any two matrices, the order of the two matrices must be the same. Asingular matrix is a matrix whose determinant is zero, while non singular matrix is the one with a non zero determinant. Under any shear, area is. It is also used in finding the inverse of the matrices, determinants of the matrices and solving a system of linear equations. AI=IA=A, If A is any square matrix and I is an identity matrix with the same order as A, then AI=IA=A, Calculate the determinant of a 2 X 2 matrix. Given that A represents a rotation through the origin, determine the angle of rotation. Find the image of the point A(1,2) after reflection in the line y = x . Matrices are given 'orders', which basically describe the size of the matrices. Linear transformations Denition 4.1 - Linear transformation . Find the vertices of the imge of S uner the transformation given by the matrix M = \( \begin{bmatrix}-1 & 2 \\2 & 1 \end{bmatrix} \) b) Sketch S and the image of S on a coordinate grid. It follows therefore that if M is a reflection in the line inclined at a, then. The numbers are put inside big brackets. From T (x, y) = (x, y) + (a, b) = (x', y'), then (7,16) = (6,-6)+(a,b) which means a=7-6 = 1 and b=16+6 = 22. However, there are some important differences that you will see in a minute. No tracking or performance measurement cookies were served with this page. Rotation is an isometric mapping and it is usually denoted by R. Therefore R means rotation of an object through an angle. Matrix multiplication, finding the determinant and inverse matrices. 3. Another example of the central and essential role maths plays in our lives. Nowthe above information can be presented in a matrix form as. The line y=0 is the x axis. View MA122_Matrix_Transformations_Notes.pdf from MA 122 at Wilfrid Laurier University. To find where the matrix M a11 a12 a13 a21 a22 a23 a31 a32 a33 ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33) maps the point Q with coordinates (x,y,z) ( x, y, z), we multiply the matrix M by the . matrix is reflected in the line y=-x changing the signs of both co-ordinates and swapping their values. When adding or subtracting one matrix from another, the corresponding elements (entities) are /added or subtracted respectively. Your body is in front of the mirror as your image is behind it. Study the following sections therefore carefully. A rigid-body is rotating around an origin point with a fixed rate. Session Overview. Matrices represents linear transformation (when a basis is given). The Matrix Fused Multiply Add (MFMA) instructions in AMD CDNA GPUs operate on a per-wavefront basis, rather than on a per-lane (thread) basis: entries of the input and output matrices are distributed over the lanes of the wavefront's vector registers. Which of the following matrices are singular matrices? A matrix can be multiplied by a constant number (scalar) or by another matrix. A matrix can do geometric transformations! In order to get the transformation of the object in the world coordinate system, you must get an objectObject3D.matrixWorld. The matrix E represents an enlargement with scale factor 0.25, centred on the origin. Multiply a matrix of order 2 X 2 by a scalar. S is a rotation through 180 about the point (2, 3). MATHEMATICS: FORM FOUR: Topic 7 - MATRICES AND TRANSFORMATIONS, STUDY NOTES FOR ORDINARY LEVEL - ALL SUBJECTS, STUDY NOTES FOR ADVANCED LEVEL - ALL SUBJECTS, Pre-Necta and Mock Exams with ANSWERS - All Regions - All Subjects, Past Papers for all Education Levels - (Necta, Mock, Pre-Necta and School Exams). These include both affine transformations (such as translation) and projective transformations. This being the case, we can only perform addition and subtraction of matrices with the same orders. Click here to read the solution to this question matrix, the point matrix is rotated 90 degrees anti-clockwise around (0, 0). the determinant is 9 - 0 = 9 hence equal to A.S.F, The transformation that maps an object (in orange) to its image (in blue) is called a, The object has same base and equal heights. 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Use the matrix method to solve the following systems of simultaneous equations. Subscribe to the latest articles from this blog directly via email. Findinverse of each of the following matrices. Find the image of B (3,4) after reflection in the line y=-x followed by another reflection in the line y=0.Draw a sketch. Includes detailed steps and try it sections for translation, reflection, rotation, and dilation. When we take the product of a matrix and a vector, we are transforming the vector. So if A has m rows and n columns, then the order of matrix is m x n. It is important to note that the order of any matrix is given by stating the number of its rows first and then the number of its columns. Learn how exactly 2x2 matrices act as transformations of the plane. In rule (iii) above, the left most element of the row is multiplied by the top most element of the column and the right most element from the row is multiplied by the bottom most element of the column and their sums are taken: The same distance from the mirror as the object, Find the image of the point D (4,2) under reflection in the x axis. A quadrilateral, Q, has the four vertices A(2, 5), B(5, 9), C(11, 9) and D(8, 5). 5. Our mission is to provide a free, world-class education to anyone, anywhere. Requested URL: byjus.com/maths/matrices-for-class-12/, User-Agent: Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/103.0.5060.114 Safari/537.36 Edg/103.0.1264.49. So our transformation matrix should look like ([0.25, 0], [0, 0.5]). Given that (x, y) = (6,-6) and (x, y) = (7,16), (a, b) =? Reflect the point (5,4) in the line y = x. Therefore, when one transformation is followed by another order is critical. In this section we will continue our study of linear transformations by considering some basic types of matrix transformations inR 2 andR 3 that have simple geometric interpretations. Find the image of the point (2, -1) under the transformation. Mrs Frog Teacher. Pre multiplication of any 2 x 1 column vector by a 2 x 2 matrix results in a 2 x 1 column vector, is thought of as a position vector that is to mean that it is representing the points with. Therefore translation vector, The Enlargement Matrix E in Enlarging Figures, Use the enlargement matrix E in enlarging figures, Find the image of the square with vertices O(0,0), A (1,0), B (1,1) and C (0,1) under the, Find the image of (6, 9) under enlargement by the matrix, Draw the image of a unit circle with center O (0,0) under. Reflection of B in the line y=-x is B'(-4,-3). Using AMD Matrix Cores. Let us start from the beginning. Great for Algebra 2, or Pre-Calculus. Part 1 Matrix Transformations Section 4.10 Print version of the lectures in MA122 Introductory Linear Algebra When multiplying by this. if one transformation doubles the area then three applications will increase the (original) area eight times (23). Point Q (-4,3) is reflected in the y axis. Solving simultaneous equations by matrix method: Now by equating the corresponding elements, the following simultaneous equations are obtained. reflection it must be on a mirror line which passses through the origin. A matrix is a rectangular array of elements, usually numbers, e.g. CBSE Class 12 Maths Notes Chapter 3 Matrices Matrix: A matrix is an ordered rectangular array of numbers or functions. Each object in the matrix is called an element (entity). For example, the \checkmark in the above matrix M M M is at position (2, 2): (2,2): (2, 2): the 2 nd 2^\text{nd} 2 nd row and 2 nd 2^\text{nd} 2 nd column. Find the matrix of a rotation of 45 clockwise about the origin, The formulae for the all of the transformation matrices can be found in the, A composite function is the result of applying more than one function to a point or set of points, Multiplication of the transformation matrices, However, the order in which the matrices is important, If the transformation represented by matrix, Another way to remember this is, starting from, The function (or matrix) furthest to the right is applied first, If a transformation, represented by the matrix, This would be the case for any number of repeated applications, Problems may involve considering patterns and sequences formed by repeated applications of a transformation, The coordinates of point(s) follow a particular pattern, The area of a shape increases/decreases by a constant factor with each application, When performing multiple transformations on a set of points, make sure you put your transformation matrices in the correct order, you can check this in an exam but sketching a diagram and checking that the transformed point ends up where it should, You may be asked to show your workings but you can still check that you have performed you matrix multiplication correctly by putting it through your GDC. The numbers or functions are called the elements or the entries of the matrix. A stretch is a transformation which enlarges all distance in a particular direction by a constant factor. Find the image of the point A (1, 2) after a reflection in the line y = x. The pre multiplier matrix is divided row wise, that is it is divided according to its rows. Using addition, subtraction, scalar multiplication, and matrix multiplication, we can transform . The Matrix to Reflect a Point P(X, Y) in the Y-Axis, Apply the matrix to reflect a point P(X, Y) in the Y-Axis. Since we need the unknown matrix B, we can solve for p and q by using equations (i) and (iii) and we solve for r and s using equations (ii) and (iv). Draw the image ABCD on the grid, A triangle T whose vertices are A (2, 3) B (5, 3) and C (4, 1 ) is mapped onto triangle T, Triangles ABC is such that A is (2, 0), B (2, 4), C (4, 4) and ABC is such that A is (0, 2), B (-4 ,10). Operations can be performed on these positions, like translate, rotate, or scale, and result in a new, transformed position. Given a matrix A, we will strive to nd a diagonal matrix to serve as the matrix B. So the image line is the line passing through (5,0) and (7,1) and it is obtained as follows; A Matrix Operator to Rotate any Point P( X, Y ) Through 90 180, 270 and 360 about the Origin, Use a matrix operator to rotate any point P( X, Y ) through 90 180, 270 and 360 about the Origin. When multiplying by this matrix. But the line y = 0 has 0 slope because it is the x axis. I n the figure above, the circle with radius 1 unit and its image with radius 3 units C, For any transformation T, any two vectors U and V and any real number t, T is said to be a linear transformation if and only if, Suppose that T is a linear transformation such that, T(U) = (1,-2), T(V) = (-3,-1) for any vectors U and V, find, (a)Since T is a linear Transformation then, If U =(2,-7) and V=(2,-3), find the matrix of linear transformation T such that T(2U)=(-4,14) and T(3V) = (6,9), 4. PDF; A note packet set to guide students through using matrices for geometric transformations. Matrix of a linear transformation: Example 5 Dene the map T :R2 R2 and the vectors v1,v2 by letting T x1 x2 = x2 x1 , v1 = 2 1 Matrices are . Write a 3x2 matrix of the quantities of items purchased over the three days . 2. andR. 04:50. Find the image of B(3,4) under reflection in the y- axis. The matrix R represents a rotation, 90 anticlockwise about the origin. The objective is to find the matrix of given transformation. The matrix representing a rotation through angle anticlockise about the origin is \( \begin{bmatrix}cos & -sin \\sin & cos \end{bmatrix} \) . The mirror is the line of symmetry between the object and the image. Now the determinant of matrix A is then defined as the difference of the product of elements in the leading diagonal and the product of the elements in the main diagonal. Definition: A transformation in a plane is a mapping which moves an object from one position to another within the plane. Find the image of B (3,4) after reflection in the line y = -x followed by another reflection in the line y = 0. The angular velocity (according to Wikipedia [1], it should be an orbital angular velocity) is a 3-vector whose direction is prependicular to the rotation plane and magnitude is the rate of rotation. 2. it has three columns and four rows. Triangle ABC is mapped onto ABC by two successive transformations, Using the same scale and axes, draw triangles ABC, the image of triangle ABC undertransformation R. Describe fully, the transformation represented by matrix R, Triangle ABC is shown on the coordinates plane below, Given that A (-6, 5) is mapped onto A (6,-4) by a shear with y- axis invariant, Draw triangle ABC, the image of triangle ABC under the shear, Determine the matrix representing this shear, Triangle A B C is mapped on to A B C by a transformation defined by the matrix, Describe fully a single transformation that maps ABC onto AB C, Determine the matrix A giving a, b, c and d as fractions. Matrices can be added, subtracted, and multiplied just like numbers. Note that both functions we obtained from matrices above were linear transformations. List of standard matrix transformations; enlargement, rotation, reflection and shear. Consider the following table showing the number of students in each stream in each form. The transformation matrix has an effect on each point of the plan. AMD Matrix Cores can be leveraged in several ways. b. The following are the common types of matrices:- Matrices of order up to 2 X 2 Add matrices of order up to 2 X 2 When adding or subtracting one matrix from another, the corresponding elements (entities) are /added or subtracted respectively. When you look at yourself in a mirror you seem to see your body behind the mirror. Matrix Transformation notes. If k is less than one 1 , it is a squish but we still call it a stretch. the rows must match in size, and the columns must match in size. If a matrix is represented in column form, then the composite transformation is performed by multiplying matrix in order from right to left side. Transformation means changing some graphics into something else by applying rules. Donate or volunteer today! When multiplying by this matrix, the x coordinate becomes the y co-ordinate and the y-ordinate becomes the x co-ordinate. By matrix method solve the following simultaneous equations: By using matrix method solve the following simultaneous equations: 1. Write a 2x1 column matrix of the unit prices of meat and bread. We can think of a matrix as a transformation of a Vector or all vectors in space. 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. e.g. This transformation matrix creates a reflection in the line y=x. Some transformations that are non-linear on an n-dimensional Euclidean space Rn can be represented as linear transformations on the n +1-dimensional space Rn+1. The order in which transformations occur can lead to different results for example a reflection in the x-axis followed by clockwise rotation of 90 is different to rotation first, followed by the reflection. Solve for x, y and z in the following matrix equation; Determine the order of each of the following matrices; 4. Matrices, plural for matrix, are surprisingly more common than you would think. Transforming a point To transform a point (x, y) by a transformation matrix , multiply the two matrices together. Find the equation of the line y = 2x + 5 after being reflected in the line y = x. The post multiplier is divided according to its columns. This being the case, we can only perform addition and subtraction of matrices with the same orders. This transformation matrix creates a reflection in the y-axis. Linear transformations are not the only ones that can be represented by matrices. Chapter 9 Matrices and Transformations 238 that This is the cost to household G if they get company 2 to deliver their milk. The following are the common types of matrices:-. Matrices that have the same number of rows as columns are called square matrices and are of particular interest. A transformation is another word for a function: it takes in some inputs (a vector) and returns some output (a transformed vector). Triangle PQR has coordinates P(-1, 4), Q(5, 4) and R(2, -1). For example, in the Cartesian X-Y plane, the matrix multiplied by an identity matrix of the same dimension, the product is the vector itself, Inv = v. rref( )A = 1 0 0 0 1 0 0 0 1 LINEAR TRANSFORMATION This system of equations can be represented in the form Ax = b. When multiplying by this matrix, the point. Use Cramers rule to solve the following simultaneous equation, 3. Order of a Matrix: If a matrix has m rows and n columns, then its order is written as m n. If a matrix has order m n, then it has mn elements. Do the same thing for y . Linear Transformations of Matrices Calculus Absolute Maxima and Minima Accumulation Function Accumulation Problems Algebraic Functions Alternating Series Antiderivatives Application of Derivatives Approximating Areas Arc Length of a Curve Arithmetic Series Average Value of a Function Calculus of Parametric Curves Candidate Test (Translations omitted as these are created by matrix addition rather than matrix multiplication). This is also known as a linear transformation from x to b because the matrix A transforms the vector x into the vector b. On the grid provided, A (1 , 2) B (7, 2) C (4, 4) D (3, 4) is a trapezium, ABCD is mapped onto ABCD by a positive quarter turn. The reflection of the point B(x,y) in the line y = -x is B'(-y,-x). Therefore the order of above matrix is each of the number s in the matrix is called . Hello and welcome to this video about using matrices to transform figures on the coordinate plane! In this video, we will cover translations dilations, reflections, and rotations. A = [ a 11 a 12 a 21 a 22 a 31 a 32]. Matrices - Geometric Transformations. Object and World Matrices (Object and World Matrices) The matrix of the object stores the transformation information of the object, which is relative to his Parent (I understand as a container). Example:- Row R and Column C If A = [1 2] Now perform, C 1 C 2 C 1 So, A = [2 1] Orthogonal matrices represent transformations that preserves length of vectors and all angles between vectors, and all transformations that preserve length and angles are orthogonal. Part Matrix Transformations Introduction to Matrix Transformations Section 6.1 Print version of the lectures in Find where (a) T maps the origin (b) T maps the point (7, 4). If k = 1 , then this transformation is really the identity i.e. The output obtained from the previous matrix is multiplied with the new coming matrix. AB is the product of matrices A and B while BA is the product of matrix B and A. The process of performing two or more transformations in order is called successive transformation e.g. If the line passes through the origin and makes an angle a with x axis in the positive direction, then its equation is y= xtan where tanis the slope of the line. Thus AB=BA=I means either A is the inverse of B or B is the inverse of A. Transcript Practice. Matrices and Transformations Questions 1. a) (i) On the grid provided, with the same scale on both axes, draw the square S whose vertices are (0, 0), (2, 0), (2,2) and (0, 2). As shown in the above figure, there is a coordinate P. You can shear it to get a new coordinate P', which can be represented in 3D matrix form as below . (1 mk) (ii) Find the coordinates and draw the image T of S under the transformation whose matrix A maps [] It involves a change of basis matrix (which will be our \(M\) matrix) and the representation of linear transformation \(T\) as a matrix under some basis (which will be our \(A\) matrix). Lets make T a transformation matrix, Then T maps points (x, y) onto image points. c. Find the point, which is mapped by T onto the point (6, 7). A new position after a transformation on is called the. The prices are 6000/= per kg of meat and 500/= per loaf of bread on each purchasing day, If M is any square matrix, that is a matrix with order mxm or nxn and Z is another matrix with the same order as m such that. Matrix A above has three (3) rows and four (4) columns. Find the determinant of each of the following matrices. Using a unit square the image of B is (1 , 0) and D is (0 , -1 ) .Therefore , the matrix of the transformation is, Show on a diagram the unit square and it image under the transformation represented by the matrix, 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. This transformation matrix creates a reflection in the x-axis. If you are doing A-level further maths, this formula is given in the formulae booklet Transformation Matrix is a matrix that transforms one vector into another vector by the process of matrix multiplication. by . To find the image of the line y = 2x + 5, we choose at least two points on it and find their images, then we use the image points to find the equation of the image line. This transformation matrix is the identity matrix. We can ask what this "linear transformation" does to all the vectors in a space. Elementary Row Transformations. . 4. The multiplication of the elements of any row or column by a non zero number. This transformation matrix rotates the point matrix 90 degrees anti-clockwise. The order of a matrix is the number of rows and columns in the matrix. A translation T maps the point (-3, 2) into (4, 3). Apply the transformation matrix A to the vectors 1 0 , 0 1 and 1 1 and you should be able to see what the transformation does by comparing the original square (the object) with its image. Matrices and Transformations. Existence of multiplicative Identity A.I = A = I.A, I is called multiplicative Identity. When a transformation takes place on a 2D plane, it is called 2D transformation. You need to be able to write down the matrix representing a rotation about any angle. Find the image of the point P(1,0) after a rotation through 90, P is on the x axis, so after rotation through 90, Find the image of the point B (4,2) after a rotation through 90, Find the image of the point (1,2) under rotation through 180, Find the image of the point (-2,1) under rotation through 270, Find the image of (1,2) after rotation of -90, Find the image of the line passing through points a (-2,3) and B(2,8) after rotation through 90, Find the image of the point (1,2) under a rotation through 180, Therefore the image of (1, 2) after rotation through 180, Find the image of the point (5,2) under rotation of 90, Therefore the image of (5,2) under rotation of 90, Consider the triangle OPQ whose vertices are (0,0), (3,1) and (3,0) respectively which is mapped into triangle OPQ by moving it 2 units in the positive x direction and 3 units in the positive y direction. Any Point P(X, Y) into P(X,Y) by Pre-Multiplying () with a Transformation Matrix T, Transform any point P(X, Y) into P(X,Y) by pre-multiplying () with a transformation matrix T. - Suppose a point P(x,y) in the x-y plane moves to a point P (x,y) by a transformation T. A transformation in which the size of the image is equal that of the object is called an ISOMETRIC MAPPING. Alsoto get r and s, the same procedure must be followed: Which of the following matrices have inverses? Cancellation Law If A is non-singular matrix, then AB = AC B = C (left cancellation law) the point matrix is rotated 90 degrees clockwise around (0, 0). Have a play with this 2D transformation app: Matrices can also transform from 3D to 2D (very useful for computer graphics), do 3D transformations and much much more. However, while we typically visualize functions with their graphs, people tend to use the word "transformation" to indicate that you should instead visualize some object moving, stretching, squishing, etc. Multiplication is done by taking an element from the row and multiplied by an element from the column. An important reason why we want to do so is that, as mentioned earlier, it allows us to compute At easily . Addition and subtraction Matrices can be added or subtracted if they have the same dimensions. Transformations can be represented by 2 X 2 matrices, and ordered pairs (coordinates) can be represented by 2 X 1 matrices. 3 Similarity Transformation to a Diagonal Matrix Henceforth, we will focus on only a special type of similarity transformation. The order is the number of rows 'by' the number of columns. This transformation matrix creates a rotation of 1 80 degrees. From the above table, if we enclose the numbers in brackets without changing To add two matrices: add the numbers in the matching positions: These are the calculations: 3+4=7. GCSE, A level, pure, mechanics, statistics, discrete if its in a Maths exam, Paul will know about it. Examples of transformations are (i) Reflection (ii) Rotation (iii) Enlargement (iv) Translation. Paul is a passionate fan of clear and colourful notes with fascinating diagrams one of the many reasons he is excited to be a member of the SME team. We are not permitting internet traffic to Byjus website from countries within European Union at this time. The red point, A on the object flag can be moved. Shear with x axis invariant is represented by a matrix of the form, Likewise a shear with y axis invariant is represented by a matrix of the form. Elementary Transformation of Matrices means playing with the rows and columns of a matrix. Describe the two transformations. Find the image of the point (1,2) after a reflection in the line y = x followed by another reflection in the line y = -x. Elementary Operation (Transformation) of a Matrix There are six operations (transformations) on a matrix, three of which are due to rows, and three are due to columns, known as elementary operations or transformations. 1. Draw a sketch. So a 2 by 3 matrix has 2 rows and 3 columns. 2 Introduction to Matrices 1.3 Addition Example showing composite transformations: The enlargement is with respect to center. Find the translation vector which maps the point (6,-6) into (7,16). Elementary transformations are those operations performed on rows and columns of the matrices to transform it into a different form so that the calculations become simpler. Multiply two matrices of order up to 2 X 2. Matrix Transformations. If T is a translation by the vector (4,3), find the image of (1, 2) under this translation. Using a unit square, find the matrix of the stretch with y axis invariant ad scale factor 3, The matrix of the stretch with the y-axis invariant and scale factor k is, with x axis invariant and scale factor k is, Isometric transformations are those in which the object and the image have the same shape and size(congruent) e.g. A house wife makes the following purchases during one week: Monday 2kg of meat and loaf of bread Wednesday, 1kg of meat and Saturday, 1kg of meat and one loaf of bread. . CSEC Math Tutor: Home Videos Add Math Mathematics SBA Past Papers Solutions CSEC Topics Ask a question Video Solutions . Visualising transformations in 3D. Matrices Matrices are tables of numbers. Matrices in Computer Graphics In OpenGL, we have multiple frames: model, world, camera frame To change frames or representation, we use transformation matrices All standard transformations (rotation, translation, scaling) can be implemented as matrix multiplications using 4x4 matrices (concatenation)

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