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# Topic: Basis (linear algebra)

 Basis (linear algebra) - Wikipedia, the free encyclopedia In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. More precisely, a basis of a vector space is a set of linearly independent vectors that span the whole space. What is called an orthonormal basis is a set of mutually orthogonal unit vectors that "span" the space via sometimes-infinite linear combinations. en.wikipedia.org /wiki/Basis_(linear_algebra)   (1187 words)

 PlanetMath: linear algebra Linear algebra is the branch of mathematics devoted to the theory of linear structure. From the geometric point of view, “linear” is synonymous with “straight”, and consequently linear algebra can be regarded as the branch of mathematics dealing with lines and planes, as well as with transformations of space that preserve “straightness”, e.g. Linearity is a very basic notion, and consequently linear algebra has applications in numerous areas of mathematics, science, and engineering. planetmath.org /encyclopedia/LinearAlgebra.html   (482 words)

 Glossary - Linear Algebra 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. A basis of a vector space is an ordered set of vectors which is linearly independent and spans the vector space. Note that every basis of a vector space has the same number of vectors, which is called the dimension of the vector space. www.math.umbc.edu /~campbell/Math221/Glossary   (1250 words)

 Centrality of Linear Algebra The separate linear algebra course became a standard part of the college mathematics curriculum in the United States in the 1950s and 60s and some colleges and universities were still adding the course in the early 1970s. Many colleges and universities are introducing a second course in linear algebra because they recognize the importance of the subject and the inadequacy of their first course in meeting the diverse needs of the students. Linear algebra provides a course that is full of ideas, with material that is rewarding to learn and to teach, and is a subject where both student and teacher can be challenged to their best performance. www.maa.org /features/cowen.html   (3738 words)

 Linear Algebra. Linear algebra is a branch of algebra which deals with linear vector spaces, linear operators as well as linear, bilinear and quadratic functions. The results of linear algebra have found application in such diverse fields as optics, quantum mechanics, display addressing, electric circuits, cryptography, computer graphics, economics, linear programming, solution of systems of differential equations, etc. The manipulation of matrices and determinants plays a central role in all applications of linear algebra. The concept of the linear vector space took shape towards the end of the nineteenth century and has dominated developments in the field of linear algebra during the twentieth century. www.ias.ac.in /currsci/jul25/articles27.htm   (1180 words)

 Linear Algebra Review Linear Algebra has become as basic and as applicable as calculus, and fortunately it is easier. In linear algebra, by convention, the column vector is viewed as the basic case. The normal linear algebra convention is that vectors are symbolized with lower-case constants/variables (a, b, c,... ling.upenn.edu /courses/Spring_2003/ling525/linear_algebra_review.html   (5601 words)

 Basis linear algebra - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-10-22) Look for Basis linear algebra in Wiktionary, our sister dictionary project. Look for Basis linear algebra in the Commons, our repository for free images, music, sound, and video. Check for Basis linear algebra in the deletion log, or visit its deletion vote page if it exists. www.sciencedaily.com /encyclopedia/basis__linear_algebra_   (165 words)

 Linear Algebra - Preliminary lectures   (Site not responding. Last check: 2007-10-22) Linear Algebra consists mostly of studying matrix calculus. Linear Algebra is then useful for solving problems in such a context with applications in topics such as Physics, Fluid Dynamics, Signal Processing and, more generally Numerical Analysis. are the coordinates of the new basis vectors (vectors of vision.unige.ch /~marchand/teaching/linalg   (1819 words)

 Linear Algebra Linear Algebra is the study of vector spaces and linear transformations. Linear transformations are functions which respect the operations of addition and scalar multiplication. Concepts from linear algebra are used in nearly every upper level mathematics course and have become quite important in physics, engineering, and statistics. www.iwu.edu /~lstout/LinearAlgebra/F02sylLinearAlgebra.html   (973 words)

 MA2102 Linear Algebra A central theme is whether a vector space has a basis consisting of eigenvectors for a given linear transformation, or equivalently, whether it has a diagonal matrix with respect to a suitable basis. The aim of this module is to introduce the student to linear algebra from a conceptual point of view, developing the introductory module MA1152. Relationship between the minimum polynomial and the existence of a basis of eigenvectors and diagonalizability of matrices. www.mcs.le.ac.uk /Modules/MA-02-03/MA2102.html   (504 words)

 Linear Algebra Some of them are obscure; mayhap even original with me. This exposition also provides documentation for the library of Linear Algebra programs that I am writing (from scratch) concurrently in the C-language, which recently became accessible to me. They already were solving simultaneous linear algebraic equations before written history. In April 1980, I was retained to write a library of Linear Algebra routines, to be written in a proprietary dialect of Fortran IV, to run on the Data General Mainframe computer. www.rism.com /LinAlg/linear_algebra.htm   (1243 words)

 Linear Dependence of Vectors. Basis of the Vector Space. It is well-known in algebra that a polynomial is identically null if all its coefficients are zeros. is not expressable as a linear combination of vectors of V. is a basis of the space iff it is the maximal linearly independent subset. www.cs.ut.ee /~toomas_l/linalg/lin1/node7.html   (377 words)

 Linear Algebra project If V is a linear space with scalars in a field K acting on the left (suppose that K is not commutative), then the dual V* is a right K-linear space (but not a left K-linear space). By linear ambient I mean a linear space that's the mother of all subspaces in a given problem. A linear ambient (feel free to suggest another name) is not a subspace, it is the space. mate.dm.uba.ar /~caniglia/Linear   (1503 words)

 A Linear Algebra View of the Wavelet Transform Linear algebra provides a tool for illuminating some wavelet algorithms and for developing wavelet and scaling function coefficients for the edges of a finite signal. Given my shallow background in linear algebra, I feel a bit unsure defining these terms, but I'll walk out on a limb (if I'm wrong, please send me e-mail with the correct explanation, preferably couched in the kind of simple terms used here). A wavelet basis function refers to the number of coefficients in the scaling and wavelet function. www.bearcave.com /misl/misl_tech/wavelets/matrix   (1660 words)

 MC241 Linear Algebra To familiarize the student with the central Mathematical notion of linear transformation, and with the properties of such transformations. At the conclusion of the course, the student should be able to decide whether a vector space has a basis of eigenvectors for a given linear transformation and appreciate how to choose a basis with respect to which the matrix of the transformation has a particularly manageable form. The student should also be able to construct an orthonormal basis for a given finite-dimensional inner product space and understand how to find an orthonormal basis of eigenvectors of a normal linear transformation. www.mcs.le.ac.uk /Modules/Year4_98-99/MC241.html   (460 words)

 MA210 Linear Algebra   (Site not responding. Last check: 2007-10-22) MA210 Linear Algebra is an introduction to vector spaces and subspaces, linear dependence and independence, basis and dimension, matrix algebra, solution of equations by matrix reduction, determinants, matrix inversion, linear transformations, eigenvalues, and eigenvectors. The course also includes applications of linear algebra and a proof component in which students learn what is needed in proofs and develop the ability to read and write proofs. * Tucker, A. Linear Algebra: An Introduction to the Theory and Use of Vectors and Matrices. www.rivier.edu /faculty/vriabov/MA210_sp04_syl.htm   (995 words)

 Linear Algebra - Content Standards (CA Dept of Education) Linear algebra is most often combined with another subject, such as trigonometry, mathematical analysis, or precalculus. For example, the solution set of a single linear equation in two variables is interpreted as a line in the plane, and the solution set of a two-by-two system is interpreted as the intersection of a pair of lines in the plane. 10.0 Students compute the determinants of 2 x 2 and 3 x 3 matrices and are familiar with their geometric interpretations as the area and volume of the parallelepipeds spanned by the images under the matrices of the standard basis vectors in two-dimensional and three-dimensional spaces. www.cde.ca.gov /be/st/ss/mthlinearalgebra.asp   (369 words)

 Linear Algebra In interpolation, the number of basis functions, n, is chosen to be equal to the number of data points, m, while for curve fitting n is typically much smaller than m. If the linear model is constructed so that it changes much more slowly than an interpolant, then curve fitting can be used as a noise filter. Now if y is not in the plane Ran(A), as is the case in Figure 1, the linear system (3) and the interpolation problem will not have a solution. www.math.clemson.edu /~bmoss/math_on_web/p2linalg1.html   (1957 words)

 Osni Marques' Home Page In this study, a linear model for the Earth structure is obtained by discretizing the Earth into layers, and the layers into cells. The velocity of wave propagation in each cell is represented by a set of parameters that account for anisotropy, location (and correction) of sources and receivers, etc. These parameters are written in matrix form and the goal is to fit the model by means of some known data. The goal is to study conformational changes of proteins by means of linear combinations of normal modes coordinates. crd.lbl.gov /~osni   (1699 words)

 MC241 Linear Algebra The course continues the study of linear algebra, taking a more conceptual point of view than MC147. The aim of this module is to introduce the student to linear algebra from a conceptual point of view, developing the introductory module MC147. Hermitian, unitary and normal transformations (with discussion of the existence of an orthonormal basis of eigenvectors for such transformations, and application to quadratic forms). www.mcs.le.ac.uk /Modules/Modules01-02/node38.html   (528 words)

 My World of Linear Algebra   (Site not responding. Last check: 2007-10-22) Applied Linear Algebra and Matrix Analysis A textbook for an introductory linear algebra course. Linear Algebra Octave Files Program files for a Matlab-like program called Octave which I have found very useful in linear algebra. Matlab is a computing standard for numerical experimentation and research, so I think that instructors ought to give serious consideration to using it or something close, like Octave, in their applied linear algebra courses, to give their students an edge up on skills they need to acquire. www.math.unl.edu /~tshores1/mylinalg.html   (816 words)

 Linear Algebra Here you will find information concerning the Linear Algebra (2270-1) class for the Spring 2003 Semester, which meets on Monday, Tuesday, Wednesday, and Friday 8:35AM-9:25AM in LCB 121. Continued discussion of the image of a linear transformation and notion of kernel. Linear combinations and product of a matrix by a vector. www.math.utah.edu /~jfernand/teaching/2270/spring03   (776 words)

 Basis   (Site not responding. Last check: 2007-10-22) Given a vector space, a basis is a maximal set of linearly independent vectors. If a linear combination of vectors drawn from any of these sets is 0, those vectors are all present in one of the sets in c. Suppose s is a set of vectors that forms a basis, and y is a vector that is not spanned by s. www.mathreference.com /la,basis.html   (500 words)

 The Math Forum - Math Library - Linear Algebra   (Site not responding. Last check: 2007-10-22) Linear algebra, sometimes disguised as matrix theory, considers sets and functions which preserve linear structure. A discussion around reform of the curriculum and the teaching of calculus and linear algebra, including differential equations, a mix of theoretical issues and concrete practices. A college algebra course to provide students with an understanding of the general concepts of relation and function and specifically of polynomial, exponential, and logarithmic functions; with the ability to solve system of equations by utilizing matrices...more>> mathforum.org /library/topics/linear   (2238 words)

 Linear Algebra   (Site not responding. Last check: 2007-10-22) Linear Algebra is the of study linear sets of equations, and the various ways you can transform them. Linear algebra has many practical applications, especially in physics and engineering. Students who take Linear Algebra must have taken or currently take A.P. Calculus or Analysis I. www.mbhs.edu /departments/magnet/coursesandlife/mathematics/linear   (57 words)

 Linear Algebra Syllabus   (Site not responding. Last check: 2007-10-22) The essential topics of linear algebra are prerequisite for many of the subsequent mathematics courses. In particular, certain linear algebra concepts are decidedly useful in multivariable calculus, differential equations and statistics. The purpose of the course is to provide an introduction to linear algebra at an elementary level to improve students' ability of abstract reasoning by its attention to mathematical proof. www.rowan.edu /math/Syllbabi/UND.COURSES/LinAlg.html   (220 words)

 Linear Algebra (Math 2318) - Vector Spaces - Basis and Dimension In this section we’re going to take a look at an important idea in the study of vector spaces.  We will also be drawing heavily on the ideas from the previous two sections and so make sure that you are comfortable with the ideas of span and linear independence. If this system has only the trivial solution the vectors will be linearly independent and if it has solutions other than the trivial solution then the vectors will be linearly dependent. Likewise, in Example 1(c) from the section on Linear Independence we saw that these vectors are linearly independent. tutorial.math.lamar.edu /AllBrowsers/2318/Basis.asp   (856 words)

 MATH-410, LINEAR ALGEBRA   (Site not responding. Last check: 2007-10-22) Vector spaces: Vectors and scalars, linear combinations, linear independence, basis, linear subspaces, direct sums. Linear operators and matrices:Matrix of a linear operator in a given basis, algebra of linear operators, change of basis, adjoint operators, range and kernel, rank and nullity, Fredholm alternative, linear functionals, self-adjoint, orthogonal, unitary, and positive definite operators, invariant subspaces and direct sums of operators, projectors. Applications in differential equations and mechanics: Homogeneous linear systems of differential equations with constant coefficients and their general solutions, small oscillations and Lagrangian mechanics. www.rpi.edu /~kovacg/classes/linearalgebra/410.html   (416 words)

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