Transformation matrices

Scale factor for the change in area and is governed by:\[\begin{bmatrix}a&0\\0&b\end{bmatrix}\] Frequently Asked Questions about Linear Transformations of Matrices Are all matrices linear transformations? Not all matrices are linear transformations- they must fit one of the linear transformation formats to be a linear transformation. What is the formula of matrices transformation? A linear transformation will have the form of ax+by and cx+dy in a matrix formation. Why are matrices linear transformations? We can use matrix multiplication to reflect a linear transformation by multiplying by a vector of x and y. Can any linear transformation be represented by matrices? Yes, any linear transformation can be represented as a matrix. What is an example of linear transformation of matrices? Reflection, rotation and enlargement/stretching are all examples of linear transformations. Save Article How we ensure our content is accurate and trustworthy? At StudySmarter, we have created a learning platform that serves millions of students. Meet the people who work hard to deliver fact based content as well as making sure it is verified. Content Creation Process: Lily Hulatt is a Digital Content Specialist with over three years of experience in content strategy and curriculum design. She gained her PhD in English Literature from Durham University in 2022, taught in Durham University’s English Studies Department, and has contributed to a number of publications. Lily specialises in English Literature, English Language, History, and Philosophy. Get to know Lily Content Quality Monitored by: Gabriel Freitas is an AI Engineer with a solid experience in software development,

Fact Checked Content Last Updated: 13.01.2023 13 min reading time Content creation process designed by Content cross-checked by Content quality checked by Sign up for free to save, edit & create flashcards. Save Article Save Article Linear Transformations of Matrices ExplanationA linear transformation is a type of transformation with certain restrictions and factors placed on it. To be a linear transformation:The origin must always stay where it was before the transformation - it is an invariant point.Transformation must be linear - no powers of \(x\) or \(y\) can be included.Transformation must be able to be described by a matrix.An invariant point or line is one that does not move during a linear transformation.Considering these factors we can then experience several types of transformations and combinations of these. The linear transformations we can use matrices to represent are:ReflectionRotationEnlargementStretchesLinear Transformations of Matrices FormulaWhen it comes to linear transformations there is a general formula that must be met for the matrix to represent a linear transformation. Any transformation must be in the form \(ax+by\). Consider the linear transformation \((T)\) of a point defined by the position vector \(\begin{bmatrix}x\\y\end{bmatrix}\). The resulting transformation could be written as this:\[T:\begin{bmatrix}x\\y\end{bmatrix}\rightarrow \begin{bmatrix}ax+by\\cx+dy\end{bmatrix}.\] Here we see \(ax+by\) and \(cx+dy\) to be describing the transformations in the \(x\) and \(y\) planes from the starting point to create our new point - the image (denoted by \(X'\) where \(X\) is the original vertex label). All we do is substitute in our values. Let's have a look at how this works.We are

Class 12 Chapter 2 Inverse Trigonometric FunctionsNCERT Exemplar Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric FunctionsChapter 3 MatricesIn Chapter 3 of NCERT textbook, we shall see the definition of a matrix, types of matrices, equality of matrices, operations on matrices such as the addition of matrices and multiplication of a matrix by a scalar, properties of matrix addition, properties of scalar multiplication, multiplication of matrices, properties of multiplication of matrices, transpose of a matrix, properties of the transpose of the matrix, symmetric and skew-symmetric matrices, elementary operation or transformation of a matrix, the inverse of a matrix by elementary operations and miscellaneous examples. Here, you can find the exercise solution links for the topics covered in this chapter.Topics Covered in Class 12 Maths Chapter 3 Matrices:Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).Maths Class 12 NCERT Solutions Chapter 3 ExercisesExercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Miscellaneous-ExerciseAlso access the following resources for Class 12 Chapter 3 Matrices at BYJU’S:CBSE Class 12 Maths Notes Chapter 3 MatricesImportant Questions for Class 12 Maths Chapter 3 – MatricesMaths Revision Notes

A B C D E F G HI JKL M N O P Q R S T U V W X Y Z N.B. An asterisk before a word means it has its own entry in the glossary.T2KA new font rasterizer - suitable for embedding in all sorts of devices - by Type Solutions' Sampo Kaasila. (He's the inventor of TrueType.) As well as doing a very good job on TrueType and Type 1 fonts without even looking at the hints, it has its own highly efficient (and hintable) outline format. Sample renderings, a downloadable demo and more information are at the Type Solutions website. In early 1998, Sun licensed T2K for use in future Java libraries.transformation matrixTwo-by-two matrices are used at several stages in TrueType fonts. First is the transformation decided by point-size, the device and the *resolution. (Remember some devices have non-square pixels.) Second is the transformation given by the current zoom ratio, and any rotations, shears or reflections. Third is the transformation associated with components of *composite *glyphs (which also have x and y offsets). Unfortunately Microsoft coded this last transformation wrongly in Windows 3.1 - only simple reflections worked. Windows 95 and NT4 got it almost right... the word is they'll soon converge back on the original Apple method. That transformation matrices are used so extensively relies on a very useful propery of *Bézier curves - that by simply transforming the control points of a curve by a certain matrix, the resulting curve is exactly as

They will be considered duplicatedReturnsNameTypeDescriptionduplicatedPartsOccurrenceListDuplicated part occurrencesidentifyInstancesIdentify parts with more than one occurrence on the scenescene.identifyInstances(2)ParametersNameTypeDefaultDescriptionminOccurrenceCountInt2Min occurrence countmakeInstanceUniqueSingularize all instances on the sub-tree of an occurrencescene.makeInstanceUnique(0)ParametersNameTypeDefaultDescriptionoccurrencesOccurrenceList0Root occurrence for the processrakeSet the same parent to all descending parts (all parts will be singularized)ParametersNameTypeDefaultDescriptionoccurrenceOccurrence0Root occurrence for the processkeepInstancesBooleanfalseIf false, the part will be singularizedremoveSymmetryMatricesRemove symmetry matrices (apply matrices on geometries on nodes under an occurrence with a symmetry matrixscene.removeSymmetryMatrices(0)ParametersNameTypeDefaultDescriptionoccurrenceOccurrence0Root occurrence for the processremoveUselessInstancesRemove instances where they are not needed (prototype referenced once, ...)scene.removeUselessInstances(0)ParametersNameTypeDefaultDescriptionoccurrenceOccurrence0Root occurrence for the processresetPartTransformSet all part transformation matrices to identity in a sub-tree, transformation will be applied to the shapesscene.resetPartTransform(0)ParametersNameTypeDefaultDescriptionrootOccurrence0Root occurrence for the processresetTransformSet all transformation matrices to identity in a sub-tree.scene.resetTransform(root, True, True, False)ParametersNameTypeDefaultDescriptionrootOccurrenceRoot occurrence for the processrecursiveBooleantrueIf False, transformation will be applied only on the root and its componentskeepInstantiationBooleantrueIf False, all occurrences will be singularizedkeepPartTransformBooleanfalseIf False, transformation will be applied to the shapes (BRepShape points or TessellatedShape vertices)selectByMaximumSizeSelect all parts meeting the criteriascene.selectByMaximumSize(roots, 150, -1, False)ParametersNameTypeDefaultDescriptionrootsOccurrenceListRoots occurrences for the processmaxDiagLengthDistance150If the diagonal axis of the bounding box is less than maxDiagLength, part will be selected. -1 to ignoremaxSizeDistance-1If the longer axis of the box is less than maxLength, part will be selected. -1 to ignoreselectHiddenBooleanfalseIf true, hidden parts meeting the criteria will be selected as wellselectDuplicatedSelect duplicated partsscene.selectDuplicated(0.01, 0.1, 0.01, 0.1)ParametersNameTypeDefaultDescriptionacceptVolumeRatioReal0.01If the ratio of volumes of two part is lower than acceptVolumeRatio, they will be considered duplicatedacceptPolycountRatioReal0.1If the ratio of polygon counts of two part is lower than acceptPolycountRatio, they will be considered duplicatedacceptAABBAxisRatioReal0.01If the ratio of AABB axis of two part is lower than acceptAABBAxisRatio, they will be considered duplicatedacceptAABBCenterDistanceDistance0.1If the ratio of AABB centers of two part is lower than acceptAABBCenterRatio, they will be considered duplicatedselectInstancesSelect occurrences sharing the same prototype as the given onescene.selectInstances(occurrence)ParametersNameTypeDefaultDescriptionoccurrenceOccurrenceReference part occurrenceselectPartsFromNoShowSelect hidden partsscene.selectPartsFromNoShow()selectVisiblePartsSelect visible partsscene.selectVisibleParts()createRayProberCreates a ray proberReturnsNameTypeDescriptionidIdentcreateSphereProberCreates a sphere proberscene.createSphereProber()ReturnsNameTypeDescriptionidIdentrayCastParametersNameTypeDefaultDescriptionrayRayThe ray to castrootOccurrenceThe root occurrence to cast fromReturnsNameTypeDescriptionhitRayHitInformation of the first ray hitrayCastAllscene.rayCastAll(ray, root)ParametersNameTypeDefaultDescriptionrayRayThe ray to castrootOccurrenceThe root occurrence to cast fromReturnsNameTypeDescriptionhitsRayHitListInformation of the first ray hitupdateRayProberUpdates the designed ray proberscene.updateRayProber(proberID, ray)ParametersNameTypeDefaultDescriptionproberIDIdentThe ray prober IdrayRayUpdate the prober's ray valuesupdateSphereProberUpdates the designed sphere proberscene.updateSphereProber(proberID, sphereCenter, sphereRadius)ParametersNameTypeDefaultDescriptionproberIDIdentThe sphere prober IdsphereCenterVector3The new prober centersphereRadiusDoubleThe new prober radiusonRayProbeParametersNameTypeDescriptionproberIDIdentThe ray propber IDproberInfoProberInfoThe prober's infoonSphereProbeParametersNameTypeDescriptionproberIDIdentThe sphere propber IDproberInfoProberInfoThe prober's infogetMultipleOccurrenceUserDataBatch version of getOccurrenceUserDatascene.getMultipleOccurrenceUserData(userDataId, occurrences)ParametersNameTypeDefaultDescriptionuserDataIdOccurrenceUserDataUserData identifier provided by subscribeToOccurrenceUserDataoccurrencesOccurrenceListOccurrences that store the user dataReturnsNameTypeDescriptionuserDataListPtrListUser data stored for each given occurrencegetOccurrenceUserDataSet or replace a userdata stored on an occurrencescene.getOccurrenceUserData(userDataId, occurrence)ParametersNameTypeDefaultDescriptionuserDataIdOccurrenceUserDataUserData identifier provided by subscribeToOccurrenceUserDataoccurrenceOccurrenceOccurrence that store the user dataReturnsNameTypeDescriptionuserDataPtrUser data stored in the given occurrencehasMultipleOccurrenceUserDataBatch version of hasOccurrenceUserDatascene.hasMultipleOccurrenceUserData(userDataId, occurrences)ParametersNameTypeDefaultDescriptionuserDataIdOccurrenceUserDataUserData identifier provided by subscribeToOccurrenceUserDataoccurrencesOccurrenceListOccurrences that potentially store the user dataReturnsNameTypeDescriptionresultsBoolListReturns an array of bool that are true if a userdata is stored on the occurrence at the same index for the given userDataIdhasOccurrenceUserDataSet or replace a userdata