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# Topic: Linear transformation

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 Affine transformation - Wikipedia, the free encyclopedia The invertible affine transformations form the affine group, which has the general linear group of degree n as subgroup and is itself a subgroup of the general linear group of degree n+1. A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations. A linear subspace of a vector space is a subset that is closed under linear combinations; an affine subspace is one that is closed under affine combinations. en.wikipedia.org /wiki/Affine_transformation   (1071 words)

 Encyclopedia: Linear transformation In the language of abstract algebra, a linear transformation is a homomorphism of vector spaces. The invertible affine transformation form the affine group, which has the general linear group of degree n as subgroup and itself is a subgroup of the general linear group of degree n+1. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. www.nationmaster.com /encyclopedia/Linear-transformation   (367 words)

 No Title   (Site not responding. Last check: 2007-10-22) There are many types of transformations that are useful in mathematics, but linear transformations are the most important. The key property for a linear transformation is that a linear transformation of a linear combination of vectors is a linear combination of the transformed vectors: You might find linear transformations difficult at first, but a type of linear transformation that you are already familiar with is a matrix. www.sci.wsu.edu /math/faculty/genz/220v/lessons/l04/l04.html   (175 words)

 Glossary - Linear Algebra   (Site not responding. Last check: 2007-10-22) A linear transformation, T, is 1-to-1 if each vector in the range of T has at most a single preimage. The linear transformation T is 1-to-1 if and only if the null space of its corresponding matrix has only the zero vector in its null space. The domain of a linear transformation is the vector space on which the transformation acts. www.math.umbc.edu /~campbell/Math221/Glossary   (1250 words)

 MTH-1A24 : Linear Algebra II A linear transformation is determined by its effect on a basis, and, given bases, we can encode all the information needed to describe a linear transformation in a matrix. The origins of linear algebra are in the middle of the 18th century when mathematicians began to analyse systems of (linear) equations more closely. Linear Transformations: Definition and examples (the linear transformation arising from a matrix; examples in analysis; geometric examples such as rotation and projection). www.mth.uea.ac.uk /maths/syllabuses/0203/1A2403.html   (584 words)

 Affine Transformations The transformations that move lines into lines, while preserving their intersection properties, are special and interesting, because they will move all polylines into polylines and all polygons into polygons. Evidently coefficients A, B, D, and E determine a linear transformation and coefficients C and F determine a parallel translation: that is, such three-by-three matrices describe affine two-dimensional transformations. Recall that, for a linear transformation, the first column of the matrix contains the coefficients of the point where the first basis vector (1, 0) is sent and the second column contains the coefficients of the point where the second basis vector (0, 1) is sent. www.quantdec.com /GIS/affine.htm   (3073 words)

 Read about Linear transformation at WorldVillage Encyclopedia. Research Linear transformation and learn about Linear ...   (Site not responding. Last check: 2007-10-22) The composition of linear transformations is linear: if f : V → W and g : W → Z are linear, then so is g o f : V → Z. If f : V → W is linear and a is an element of the ground field K, then the map af, defined by (af)(x) = a (f(x)), is also linear. matrices, the addition of linear maps corresponds to the addition of matrices, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars. encyclopedia.worldvillage.com /s/b/Linear_operator   (893 words)

 Math 310 - Glossary of Linear Algebra Terms   (Site not responding. Last check: 2007-10-22) The nullity of a linear transformation is the dimension of its null space. A linear transformation T from V to W is orthogonal if T(v) has the same length as v for all vectors v in V. The rank of a linear transformation (and hence of any matrix regarded as a linear transformation) is the dimension of its range. www.math.uic.edu /math310/glossary.html   (1060 words)

 Linear Estimators   (Site not responding. Last check: 2007-10-22) Because of the transformation's linearity, the theory of linear vector spaces can be fruitfully used to derive the estimator and to specify its properties. The linear estimator is that particular linear transformation that yields an estimation error orthogonal to all linear transformations of the data. Linear transformations, however, are homogeneous: A zero-values input yields a zero output. cnx.rice.edu /content/m11276/latest   (1052 words)

 Linear Algebra (Math 2318) - Euclidean n-Space - Linear Transformations   (Site not responding. Last check: 2007-10-22) Now, each of the components are given by a system of linear (hhmm, makes one instantly wonder if the transformation is also linear…) equations and we saw in the first chapter that we can always write a system of linear equations in matrix form.  Let’s do that for this system. There are a couple of things to note here.  First, we couldn’t write the transformation from Example 2 as a matrix multiplication because at least one of the equations (okay both in this case) for the components in the result were not linear. The other thing that we saw in Example 4 is that we were able, in that case, to write a linear transformation as a matrix multiplication.  Again, it turns out that every linear transformation can be written as a matrix multiplication. tutorial.math.lamar.edu /AllBrowsers/2318/LinearTransformations.asp   (1153 words)

 Sociology: UTSA Style: Methods and Measurements: Data Manipulation Techniques. Transformations for normality in ... When a Z transformation is done on a normal population, the resulting distribution is called the Standard Normal Distribution or the Z-Distribution. Linear transformations attempt to translate the data to a new origin (think in terms of a number line) by expanding (or contracting) the scale of measurement. This may not be the case and it may be necessary to transform the data to normality by a non-linear technique to meet the assumptions imposed by various statistical techniques. colfa.utsa.edu /Sociology/masters/manip.html   (1335 words)

 Linear Transformations, Part 2 A transformation T mapping V to W is called surjective (or onto) if every vector w in W is the image of some vector v in V. A linear transformation T from a vector space V to a vector space W is called an isomorphism of vector spaces if T is both injective and surjective. If a linear transformation is an isomorphism and is defined by multiplication by a matrix, explain why the matrix must be square. www.math.duke.edu /education/ccp/materials/linalg/lintrans/lint2.html   (332 words)

 Linear transformations Criterion for the linearity of a transformation of V Matrix of a linear transformation with respect to a basis in V. Null-space of a linear transformation It can easily be proved that t+t' is a linear transformation and that the matrix of t+t' is equal to the sum of the matrices of t and of t'. www.cartage.org.lb /en/themes/Sciences/Mathematics/Algebra/foci/topics/transformations/transformations.htm   (2070 words)

 Mobius Transformations This family of transformations is also called fractional linear transformations. Composition of Mobius transformations is a Mobius transformation. Linear transformations are composed of shifts, rotations and scale multiplications, so they transform lines to lines and circles to circles. www.geocities.com /assafwool/Mobius/Mobius.html   (614 words)

 LINEAR TRANSFORMATIONS The central objective of linear algebra is the analysis of linear functions defined on a finite-dimensional vector space. In example 14 the kernel of the given linear transformation was a subspace of the domain. Thus for the linear transformation in Example 14 the nullity is 1. distance-ed.math.tamu.edu /Math640/chapter4/node2.html   (880 words)

 Conformal Mappings   (Site not responding. Last check: 2007-10-22) is called a linear transformation and it is a one-to-one mapping of the complex z-plane onto the complex w-plane. Since a linear transformation can be considered as a composition of a rotation, a magnification, and a translation, it follows that a linear transformations preserve angles. These mappings are conveniently expressed as the quotient of two linear expressions and are commonly known as linear fractional or bilinear transformations. mathews.ecs.fullerton.edu /fofz/conformal/c0.htm   (444 words)

 Creating a Gallery of Transformed Images (Image Processing Toolbox Image Transformation Demos)   (Site not responding. Last check: 2007-10-22) Linear conformal transformations may include a rotation, a scaling, and a translation. Linear conformal transformations are a subset of affine transformations. In a piecewise linear transformation, linear transformations are applied separately to different parts of the image. www.clemson.edu /cle4_share/CWE/COES0915_CLUG/REFERENCE/matlabr14/toolbox/images/imdemos/html/tform.html   (896 words)

 Linear Transformations If you want postmultiplication to be a linear map, our vector space is more than a left r module, it is a bimodule, with r on either side. Just to confuse you, a linear transformation is also called a linear function, or a linear map, or a linear operator, or a linear transform. If a linear combination of elements in w becomes 0, then the same linear combination of preimages from y maps to something in the kernel, which is spanned by x. www.mathreference.com /la,xform.html   (959 words)

 Algebra:Linear transformations - Wikibooks, collection of open-content textbooks If we can find just one case in which the transformation does not preserve addition, scalar multiplication, or the zero vector, we can conclude that the transformation is not linear. We have some fundamental concepts underlying linear transformations, such as the image and the kernel of a linear transformation, which are analogous to the range and zeros of a function. The image of a linear transformation T:V->W is the set of all vectors in W which were mapped from vectors in V. For example with the trivial mapping T:V->W such that Tx=0, the image would be 0. en.wikibooks.org /wiki/Algebra:Linear_transformations   (886 words)

 Linear transformation There are also important examples of linear transformation involving infinite-dimensional spaces. For instance, the integral yields a linear map from the space of all real-valued integrable functions on some interval to R, while differentiation is a linear transformation from the space of all differentiable functions to the space of all functions. In the finite dimensional case and if bases have been chosen, then the composition of linear maps corresponds to the multiplication of matrices, the addition of linear maps corresponds ot the addition of matrices, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars. www.brainyencyclopedia.com /encyclopedia/l/li/linear_transformation.html   (713 words)

 Margaret Sinclair: Linear transformations using spreadsheets This unit investigates the effect of a variety of linear transformations on a unit square and a rectangle. Students record their observations in a table and conjecture relationships between characteristics of the transformation matrix and the shape, orientation and area of the image. If the linear transformation is an isometry, then the lengths of vectors and angles betweeen vectors remain unchanged under the transformation. mathforum.org /workshops/sum98/participants/sinclair/outline.html   (526 words)

 [No title] W, is a linear transformation if The linear transformation T:V -> W, is non-singular if The linear operator, T:V -> V, has characteristic value c if Given an n by n matrix A with coefficients in F, the minimal polynomial of A is Suppose A and B are n by n matrices. Suppose T is a linear operator on a finite dimensional vector space V over the field F. Prove: If r is a root for the characteristic polynomial of T then r is a root of the minimal polynomial of T. Suppose A and B are nxn matrices. Since the chairperson noticed you were just completing an upper division course in linear algebra at HSU and no one else in the department has had a linear algebra course in 20 years (if they had one at all), your specific team assignment is to cover the topics related to transformations. www.humboldt.edu /~mef2/Courses/M344SampleFinalProblems.doc   (935 words)

 Geometry of Linear Transformations - HMC Calculus Tutorial Since each linear transformation of the plane has a unique standard matrix, we will identify linear transformations of the plane by their standard matrices. For every line in the plane, there is a linear transformation that reflects vectors about that line. Every linear transformation of the plane with an invertible standard matrix has the geometric effect of a sequence of reflections, expansions, compressions, and shears. www.math.hmc.edu /calculus/tutorials/lineartransformations   (346 words)

 Multiplication of a Vector by a Matrix We see that a composition g(f(x)) of two linear transformations is in turn linear. What we arrived at is that a linear transformation of a vector space may be expressed as a product of a matrix and a vector. To obtain the result of applying a transformation to a vector multiply the corresponding matrix and the tuple. www.cut-the-knot.org /do_you_know/mul_vm.shtml   (429 words)

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