
Review of Short Phrases and Links 
This Review contains major "Linear System" related terms, short phrases and links grouped together in the form of Encyclopedia article.
Definitions
 A linear system is a mathematical model of a system based on the use of a linear operator.
 The linear system is a projective space, so we've got the Zariski topology on it.
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 A linear system is called inconsistent or overdetermined if it does not have a solution.
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 A linear system is a set of linear equations and a homogeneous linear system is a set of homogeneous linear equations.
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 The equations of a linear system are independent if none of the equations can be derived algebraically from the others.
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 SSPSL solves a linear system factored by SSPFA or SSPCO. SSVDC computes the singular value decomposition of a general matrix.
 Objective: Solve a 3x3 linear system of equations.
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 DPOSL solves a linear system factored by DPOCO or DPOFA. DPPCO factors a double precision symmetric positive definite matrix in packed form.
 L eq K: LinSys,LinSys  BoolElt True if and only if the linear systems L and K are equal if considered as linear subsystems of some complete linear system.
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 SGESL solves a linear system factored by SGECO or SGEFA. SGTSL solves a linear system, unfactored tridiagonal matrix.
 SPPSL solves a linear system factored by SPPCO or SPPFA. SPTSL solves a linear system for a positive definite tridiagonal matrix.
 SGBSL solves a linear system involving a general band matrix.
 Gauss elimination on a linear system is an example of such a direct method.
 SPBSL solves a linear system factored by SPBCO or SPBFA. SPOCO factors a positive definite matrix and estimates its condition number.
 SSISL solves a linear system factored by SSIFA or SSICO. SSPCO factors a symmetric indefinite packed matrix and estimates its condition number.
 SPOSL solves a linear system factored by SPOCO or SPOFA. SPPCO factors a positive definite packed matrix and estimates its condition number.
 Probably the first iterative method for solving a linear system appeared in a letter of Gauss to a student of his.
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 Because these operations are reversible, the augmented matrix produced always represents a linear system that is equivalent to the original.
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 Since Krylov methods can suffer from slow convergence, one can modify the original linear system in order to improve convergence properties.
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 A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied.
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 A linear system in three variables determines a collection of planes.
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 The incomplete LU factorization is intended as a preconditioner, which modifies the linear system, improving the convergence rate of the iterative scheme.
 These routines can compute the LU factorization, determinant, inverse, or solution of a linear system.
 Initially, we present three methods by which a linear system can be created.
 Alternatively, a sequence of monomials of some common degree can be specified to generate the sections of a linear system.
 Typically, the complete linear system of all hypersurfaces of a given degree is created and then restricted by the imposition of geometrical conditions.
 Multiplicity(L,p) : LinearSys,Pt  RngIntElt The generic multiplicity of hypersurfaces of the linear system L at the point p.
 The intrinsic Reduction creates a new linear system by removing this codimension 1 base locus, as is seen below.
 BasePoints(L) : LinearSys  SeqEnum A sequence containing the basepoints of the linear system L if the base locus of L is finite dimensional.
 BasePoints(L): LinearSys  SeqEnum A sequence containing the basepoints of the linear system L if the base locus of L is finite dimensional.
 The cgmethod performs an orthogonal projection of the original preconditioned linear system to another system of smaller dimension.
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 See also Linear system of divisors in algebraic geometry.
 A general linear system corresponds to some vector subspace of the coefficient space of a complete linear system.
 Most cases (but not all!) requiring the solution of a linear system involve a square coefficient matrix.
 The QR factorization transforms an overdetermined linear system into an equivalent triangular system.
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 It can be difficult to describe the solution set to a linear system with infinitely many solutions.
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 In other words, the linear system defined by the sections of L after common factors are removed.
 The equations of a linear system are consistent if they possess a common solution, and inconsistent otherwise.
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 However, this method of producing an approximate solution of an overdetermined (and usually inconsistent) linear system is usually not recommended.
 Linear system dialog, ask questions to get information in order to solve linear systems.
 Because these operations are reversible, the augmented matrix produced always represents a linear system that is equivalent to the original.
 It can be difficult to describe the solution set to a linear system with infinitely many solutions.
 Typically, the matrix A is large and sparse, and an iterative scheme is being used to solve the linear system A*x=b.
 Solve linear system Mx=V, return solution V if there is a unique solution, null otherwise.
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 Deductio linear system, exercises of interactive deduction on linear systems.
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 Cyclic reduction is a method for solving a linear system A*x=b in the special case where A is a tridiagonal matrix.
 Solve the given linear system of m equations in n unknowns.
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 This linear system is known as the normal equations.
 Other (more elegant) approaches that reduce to a linear system.
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 We start by defining a linear system whose chosen sections are clearly not linearly independent.
 The complete linear system of degree d whose sections are all monomials of that degree has a special creation function.
 IsComplete(L) : LinSys  BoolElt True if and only if the linear system L is the complete linear system of polynomials of some degree.
 For example, if L is a linear system on some projective space P then the corresponding map can be created as follows.
 Pullback(f,L) : MapSch,LinearSys  LinearSys The linear system f^ * L on the domain of the map of schemes f where L is a linear system on the codomain of f.
 In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.
 Many illposed problems are solved using a discretization that results in a least squares problem or a linear system involving a Toeplitz matrix.
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 On the other hand, v is of order 3 so it imposes more conditions on the linear system.
 Solve the following linear system of equations by hand using Gaussian elimination and backsubstitution.
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 The resulting plot can then be used to pick off the response of any linear system, given its natural frequency of oscillation.
 Give an example of a linear system of equations for which you can't use Cramer's rule, and explain why Cramer's rule doesn't work for your example.
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 Finding Eigenvalues For any linear system, the key to understanding how it behaves over time is through eigenvalues and eigenvectors.
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 For the eigenvector associated with λ 1 = 3, we consider the resulting linear system.
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 Then interpret this row reduced matrix as a homogeneous linear system and find as many linearly independent solution vectors as possible.
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 If a solution to a homogeneous linear system exists at all, then the zero solution will be a solution.
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 Pullback(f,L): AmbProjMap,LinSys  LinSys The linear system f^ * L on the domain of the map of schemes f where L is a linear system on the codomain of f.
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 Linear systems in projective space are simply collections of hypersurfaces having a common degree which are parametrised linearly by a vector space.
 This leads to a quadratic minimization problem, and hence a linear system of equations (normal equations) is obtained for the unknown vector a.
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 The solution vector x of an n by n linear system A x = b is guaranteed to exist and to be unique if the coefficient array A is invertible.
 If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set.
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 I would recommend this book as a reference for introductory treatment of Linear System Theory and Design.esign.
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 The condition number is a measure of stability or sensitivity of a matrix (or the linear system it represents) to numerical operations.
 Notes The flag can be used to eliminate unnecessary work in the preconditioner during the repeated solution of linear systems of the same size.
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 Solves the linear system of equations a * x = b by means of the Preconditioned Conjugate Gradient iterative method.
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 A calculator that can solve a linear system of 3 equations and 3 unknowns, includes explanation on the mathematics behind it.
 A linear system of equations with n unknowns must have at least n equations to get a unique solution.
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 See also: linear element, linear system, nonlinearity.
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 In the case of a linear system which is underdetermined, or an otherwise non invertible matrix, singular value decomposition (SVD) is equally useful.
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 In the meantime we will satisfy ourselves by noting the advantage that matrix multiplication gives us by representing a linear system in matrix form.
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 A matrix form of a linear system of equations obtained from the coefficient matrix as shown below.
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 Numerical operations such as linear system solving and eigenvalue calculations can be applied to two different kinds of matrix: dense and sparse.
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 A solution x then can be found using back substitution, i.e., solving a linear system with an upper triangular matrix.
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 Reduced row echelon form of a linear system obtained by elementary row operations.
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 Like the earlier package, ScaLAPACK targets linear system solution and eigenvalue calculation for dense and banded matrices.
 The QR factorization is used to solve linear systems with more equations than unknowns.
 If row interchanges are not needed to solve the linear system AX = B, then A has the LU factorization (illustrated with 4×4 matrices).
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 Occasionally, a nonlinear system can be solved exactly by a change of variables to a linear system.
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Linear System
 Many illposed problems are solved using a discretization that results in a least squares problem or a linear system involving a Toeplitz matrix.
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