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Topic: Basis transformation matrix

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	PlanetMath: matrix (Site not responding. Last check: 2007-10-07)
		As a rule, though, the word matrix and the notation associated with it are used only in connection with linear mappings.
		One more illustration: The derivative (at a point) of a mapping from one surface to another is a linear mapping; it is not a matrix of partial derivatives, because the matrix depends on a choice of basis but the derivative does not.
		This is version 16 of matrix, born on 2002-02-22, modified 2006-10-27.
	planetmath.org /encyclopedia/Matrix.html (574 words)

	Matrix (mathematics) - Wikipedia, the free encyclopedia
		The entry of a matrix A that lies in the i -th row and the j-th column is called the i,j entry or (i,j)-th entry of A.
		A 1 × n matrix (one row and n columns) is called a row vector, and an m × 1 matrix (one column and m rows) is called a column vector.
		The rank of a matrix A is the dimension of the image of the linear map represented by A; this is the same as the dimension of the space generated by the rows of A, and also the same as the dimension of the space generated by the columns of A.
	en.wikipedia.org /wiki/Matrix_(mathematics) (1664 words)

	PlanetMath: nilpotent transformation (Site not responding. Last check: 2007-10-07)
		A nilpotent transformation naturally determines a flag of subspaces
		Cross-references: size, block, matrix, basis, transformation, natural numbers, sequence, linear isomorphism, signature, subspaces, flag, linear transformation
		This is version 4 of nilpotent transformation, born on 2002-02-15, modified 2004-05-04.
	planetmath.org /encyclopedia/Nilpotent2.html (93 words)

	math lessons - Linear transformation
		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.
		In the language of abstract algebra, a linear transformation is a homomorphism of vector spaces.
		Differentiation is a linear transformation from the space of all differentiable functions to the space of all functions.
	www.mathdaily.com /lessons/Linear_transformation (975 words)

	Glossary - Linear Algebra
		The column space of a matrix is the subspace of the codomain which is spanned by the columns of the matrix.
		The rank of a matrix is the dimension of the row space, which is equal to the dimension of the column space.
		The row space of a matrix is the subspace of the domain which is spanned by the rows of the matrix.
	www.math.umbc.edu /~campbell/Math221/Glossary (1250 words)

	Arithmetic Operators (Site not responding. Last check: 2007-10-07)
		Operator Op is first placed in the default basis, the matrix mx is then subtracted from it, and then the result is new operator in the default basis.
		For the subtraction of a matrix from an operator, an operator which has the matrix subtracted is returned in the default basis..
		For the equating of an operator to a matrix, Op = mx, the matrix mx is taken to represent the operator Op in the default basis.
	gamma.magnet.fsu.edu /html/modules/hslib/operato1.htm (2733 words)

	[ hacked by deviance ] - Matrix Operations
		If the matrix for the position of the car is A, the matrix of the person in the car is B, the matrix for the nodding-transformation is C, and one vector in the head of the person is v, you will first have to calculate C*v resulting in the new head position.
		If you want to transform matrices from a basis a = (a1, a2, a3) to a basis b = (b1, b2, b3) (notice that a1, a2, a3, b1, b2 and b3 are vectors) you have to express the vectors of a by means of the basis b.
		If you have the special case that the destination basis is the canonical basis E = (e1, e2, e3), which is often the case (except the camera thing), you can simply use the vectors of the source basis as the columns of the transformation matrix.
	www.cfxweb.net /modules.php?name=News&file=article&sid=163 (1843 words)

	Invertible matrix Summary
		That is a matrix with all of its (i,j) entries equal to zero whenever i > j.
		Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.
		Writing another special matrix of cofactors, known as an adjugate matrix, can also be an efficient way to calculate the inverse of small matrices (since this method is essentially recursive, it becomes inefficient for large matrices).
	www.bookrags.com /Invertible_matrix (1462 words)

	Matrix theory Summary
		Matrix mechanics was originally formulated by a group of physicists in the 1920s, most prominently by Werner Heisenberg, Max Born, and their colleagues.
		Matrix theory is a branch of mathematics which focuses on the study of matrices.
		A permutation matrix is the matrix representation of a permutation; it is a square matrix with entries 0 and 1, with just one entry 1 in each row and each column.
	www.bookrags.com /Matrix_theory (656 words)

	The Classical Coupled Mass Problem
		The solution was to write the differential equations in matrix form, and then diagonalize the matrix to obtain the eigenvectors and eigenvalues.
		Since this 2x2 matrix is real and symmetric, it must also be Hermitian, so we know that it has real eigenvalues, and that the eigenvectors will be linearly independent and can be made to form an orthonormal basis.
		The matrix equation 13 is of course equivalent to the two simple equations
	vergil.chemistry.gatech.edu /notes/time_evolution/node2.html (635 words)

	Matrix Question - Basis Vectors - GameDev.Net Discussion Forums
		Assuming that all other elements resemble the identity matrix (i.e anything besides the translation deltas are 1 or 0 accordingly), a matrix with the translation stored in the last column is the transpose of the matrix that has the translation stored in the bottom row.
		When homogeneization occurs (after the matrix transformations), the x and y coordinates are both divided by w, so the farther the object is, the greater the w, the smaller it will be on the screen (far objects appear smaller than close objects).
		So when you use a matrix to represent a point, we have three variables that represent our points position, but if this same matrix were to be used to represent an equation, we need a constant term, and with three variables, this leads to a fourth term, which is the W component.
	www.gamedev.net /community/forums/viewreply.asp?ID=643962 (4456 words)

	CSC418 - Lecture Topics - Modeling & Transformation
		Transformations are invertible, either by inverting the matrix or by undoing the transformations.
		The transformations necessary to convert a point from CS 2 to CS 1 are given by the transformations necessary to align the CS 1 frame with the CS 2 frame.
		Transformations are applied to points by augmenting the three-dimensional coordinate with a fourth homogeneous parameter, h=1.
	www.cs.toronto.edu /~faisal/teaching/notes/csc418/faisal/topics04.html (1121 words)

	Matrix Decompositions
		The LU decomposition of a matrix is frequently used as part of a Gaussian elimination process for solving a matrix equation.
		The first element is a combination of upper and lower triangular matrices, the second element is a vector specifying rows used for pivoting (a permutation vector which is equivalent to the permutation matrix), and the third element is an estimate of the condition number.
		This matrix cannot be diagonalized because the matrix of eigenvectors is singular.
	documents.wolfram.com /v5/Built-inFunctions/AdvancedDocumentation/LinearAlgebra/4.5.html (2005 words)

	Matrix Reference Manual: Special Matrices
		A 3#3 orthogonal matrix is either a rotation matrix or else a rotation matrix plus a reflection in the plane of the rotation according to whether it is proper or improper.
		A Rotation matrix is orthogonal with a determinant of +1.
		A signature matrix is a diagonal matrix whose diagonal entries are all +1 or -1.
	www.ee.ic.ac.uk /hp/staff/dmb/matrix/special.html (4464 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)

	How to find the correct transformation
		describes the transformation of the row of basis vectors (a, b, c) to the reference basis vectors (a', b', c').
		The transformation matrix pair (P,p) of a group-subgroup chain G > H used by the programs on Bilbao Crystallographic Server always describes the transformation from the reference (default) coordinate system of the group G to that of the subgroup H. Let (a,b,c)
		The second column of the rotational part of the transformation matrix is (0 0 1).
	www.cryst.ehu.es /cryst/trmatrix.html (312 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.
		maps the basis vectors to the vectors represented by the columns of the matrix.
	mathforum.org /workshops/sum98/participants/sinclair/outline.html (541 words)

	OpenGL FAQ / 9 Transformations
		The root cause of the problem is that OpenGL matrix operations postmultiply onto the matrix stack, thus causing transformations to occur in object space.
		The upper left 3x3 portion of a transformation matrix is composed of the new X, Y, and Z axes of the post-transformation coordinate space.
		Then simply construct your transformation matrix by inserting the new local X, Y, and Z axes into the upper left 3x3 portion of an identity matrix.
	www.opengl.org /resources/faq/technical/transformations.htm (3501 words)

	Maths - Matrix algebra and 3D transformation - Martin Baker
		A matrix is a rectangular array of elements which are operated on as a single object.
		As a case of a tensor (rank 2 tensor is a matrix).
		A matrix is a way to transform one vector into another vector (and a whole set of vectors into another set of vectors).
	www.euclideanspace.com /maths/algebra/matrix/index.htm (730 words)

	Matrix Market: Glossary
		A defective matrix cannot be transformed to a diagonal matrix using similarity transformations.
		A matrix is diagonally dominant if the absolute value of each diagonal element is greater than the sum of the absolute values of the other elements in its row (or column).
		A Hessenberg matrix is `almost' triangular, that is, it is (upper or lower) triangular with one additional off-diagonal band (immediately adjacent to the main diagonal).
	math.nist.gov /MatrixMarket/glossary.html (756 words)

	Evaluation of the rotation matrices in the basis of real spherical harmonics. (Site not responding. Last check: 2007-10-07)
		Rotation matrices (or Wigner D functions) are the matrix representations of the rotation operators in the basis of the spherical harmonics.
		The aim of this contribution is to obtain a general algorithm to compute the representation matrix of any point-group symmetry operation in the basis of the real spherical harmonics, paying attention to the use of recurrence relations that allow the treatment of functions with high angular momenta.
		Usually, point-group symmetry operations are specified by their matrix representations in the basis of the cartesian coordinates, so we will focus in the obtaining of the representation matrices of the RSH taking those of the cartesian coordinates as a starting point.
	www.elsevier.com /homepage/saa/eccc3/paper48/eccc3.html (2838 words)

	class HT_Matrix
		Appends a basis to the transformation represented by this HT_Matrix object.
		Appends a shear to the transformation represented by this HT_Matrix object.
		The adjoint matrix of a non-singular matrix is its inverse multiplied by its determinant.
	www.autodesk.com /heididg/AppendixA/Classes/HT_Matrix.html (783 words)

	Lattices from Matrix Groups
		Return the action of the finite rational matrix group G on an invariant lattice as an integral matrix group, thus giving an equivalent integral group H, together with the transformation matrix T from the standard lattice to the invariant lattice.
		For a rational matrix group G or a G-lattice L, return a basis for the space of invariant bilinear forms for G (represented by their Gram matrices) as a sequence of matrices.
		For a rational matrix group G or a G-lattice L, return the dimension of the space of (symmetric and anti-symmetric) invariant bilinear forms for G. The algorithm uses a modular method which is always correct and is faster than the actual computation of the forms.
	www.umich.edu /~gpcc/scs/magma/text816.htm (1767 words)

	Innovative Teaching Exchange: In Search of the Elusive Matrix
		I decided that the activity had to employ a variety of tasks [3], to probe the connections between symbols, symbolic procedures and problem-solving procedures [3], to be diverse in terms of complexity and dissimilarity from those presented in the learning situation [1], and to cause students to build connections, fold-back, or validate thinking [5].
		In particular, the activity was designed to help students coordinate their understandings of dimension, basis, null space, and column space by taking a new perspective different from that presented in David Lay's Linear Algebra and It's Applications text [4], class, or homework.
		Students were generally able to situate the concepts of null and column spaces, basis, and dimension to provide reasonably coherent discussions of the ideas.
	www.maa.org /t_and_l/exchange/ite4/insearch.html (1491 words)

	An Introduction to Wavelets: Wavelet Analysis
		The DWT matrix is not sparse in general, so we face the same complexity issues that we had previously faced for the discrete Fourier transform (7).
		The wavelet transform is actually a subset of a far more versatile transform, the wavelet packet transform (8).
		The chosen basis carries substantial information about the signal, and if the basis description is efficient (that is, very few terms in the expansion are needed to represent the signal), then that signal information has been compressed.
	www.amara.com /IEEEwave/IW_wave_ana.html (902 words)

	More on Matrices (Site not responding. Last check: 2007-10-07)
		It means that instead of expressing a vector in terms of its components along the unit vectors i=(100), j=(010), and k=(001), we express the same vector in terms of its components (or projections) along a new set of basis vectors i’, j’, and k’.
		(The term basis, and basis vectors, refers to a set of vectors that can be linearly combined, using scalar multiplication and vector addition, to point to any vector in the vector space.
		Therefore the matrix associated with a rotation of the coordinate axes by θ is
	www.udel.edu /Biology/rosewc/msp_biomech05f/notes/matrices2_mp.html (2085 words)

	Matrix tutorial
		The result of the matrix multiplication is a matrix consisting of one cell.
		All matrix multiplication can be thought of as rows on the left multiplying be columns on the right.
		When you start with an integer matrix, value of its determinant is the lowest common denominator of the elements in its inverse.
	www-users.cs.umn.edu /~arashid/cached/MatrixTutorial.htm (2118 words)

	SIMILAR MATRICES AND CHANGE OF BASIS
		We recall from matrix algebra that diagonal matrices are easy to work with for certain operations: inversion, determinants, and multiplication, to name three.
		is a transition matrix from one basis to another, as we saw in Theorem 4.3.1.
		The point is that the transition matrix is just the matrix which represents the identity transformation.
	distance-ed.math.tamu.edu /Math640/chapter4/node6.html (392 words)

	MTH-2C13: Linear Algebra II (Site not responding. Last check: 2007-10-07)
		We distinguish certain subsets of a vector space which can themselves be considered as vector spaces (subspaces), and relate the dimensions of subspaces to the dimension of the ambient space.
		Linear transformations are determined by their effect on a basis and so can be described by a matrix.
		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/0405/2C1304.html (361 words)

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