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Augmented Matrix
In linear algebra, an augmented matrix (A \vert B) is a k \times (n+1) matrix obtained by appending a k-dimensional column vector B, on the right, as a further column to a k \times n-dimensional matrix A. This is usually done for the purpose of performing the same elementary row operations on the augmented matrix (A \vert B) as is done on the original one A when solving a system of linear equations by Gaussian elimination. For example, given the matrices A and column vector B, where A = \begin 1 & 3 & 2 \\ 2 & 0 & 1 \\ 5 & 2 & 2 \end , \quad B = \begin 4 \\ 3 \\ 1 \end, the augmented matrix (A \vert B) is (A, B) = \left begin 1 & 3 & 2 & 4 \\ 2 & 0 & 1 & 3 \\ 5 & 2 & 2 & 1 \end\right For a given number n of unknowns, the number of solutions to a system of k linear equations depends only on the rank of the matrix of coefficients A representing the system and the rank of the corresponding augmented matrix (A \vert B) where the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathematics), matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as line (geometry), lines, plane (geometry), planes and rotation (mathematics), rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to Space of functions, function spaces. Linear algebra is also used in most sciences and fields of engineering because it allows mathematical model, modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Matrix (mathematics)
In mathematics, a matrix (: matrices) is a rectangle, rectangular array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a " matrix", or a matrix of dimension . Matrices are commonly used in linear algebra, where they represent linear maps. In geometry, matrices are widely used for specifying and representing geometric transformations (for example rotation (mathematics), rotations) and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Elementary Row Operations
In mathematics, an elementary matrix is a square matrix obtained from the application of a single elementary row operation to the identity matrix. The elementary matrices generate the general linear group when is a field. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations. Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form. They are also used in Gauss–Jordan elimination to further reduce the matrix to reduced row echelon form. Elementary row operations There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations): ;Row switching: A row within the matrix can be switched with another row. : R_i \leftrightarrow R_j ;Row multiplication: Each element in a row can be multiplied by a non-zero constant. It is also known as ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Gaussian Elimination
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of an invertible matrix. The method is named after Carl Friedrich Gauss (1777–1855). To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations: * Swapping two rows, * Multiplying a row by a nonzero number, * Adding a multiple of one row to another row. Using these operations, a matrix can always be transformed into an upper triangular matrix (possibly bordered by rows or columns of zeros), and in fact one that is in row echelon form. Once all of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Rank (linear Algebra)
In linear algebra, the rank of a matrix is the dimension of the vector space generated (or spanned) by its columns. p. 48, § 1.16 This corresponds to the maximal number of linearly independent columns of . This, in turn, is identical to the dimension of the vector space spanned by its rows. Rank is thus a measure of the " nondegenerateness" of the system of linear equations and linear transformation encoded by . There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics. The rank is commonly denoted by or ; sometimes the parentheses are not written, as in .Alternative notation includes \rho (\Phi) from and . Main definitions In this section, we give some definitions of the rank of a matrix. Many definitions are possible; see Alternative definitions for several of these. The column rank of is the dimension of the column space of , while the row rank of is the dimension of the row space of . A fundamental resul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Rouché–Capelli Theorem
Rouché–Capelli theorem is a theorem in linear algebra that determines the number of solutions of a system of linear equations, given the ranks of its augmented matrix and coefficient matrix. The theorem is variously known as the: * Rouché–Capelli theorem in English speaking countries, Italy and Brazil; * Kronecker–Capelli theorem in Austria, Poland, Ukraine, Croatia, Romania, Serbia and Russia; * Rouché–Fontené theorem in France; * Rouché–Frobenius theorem in Spain and many countries in Latin America; * Frobenius theorem in the Czech Republic and in Slovakia. Statement A system of linear equations with variables and coefficients in a field has a solution if and only if its coefficient matrix and its augmented matrix have the same rank. If there are solutions, they form an affine subspace of K^n of dimension . In particular: * if , the solution is unique, * if and is an infinite field, the system of linear equations admits infinitely many solutions, * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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System Of Linear Equations
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variable (math), variables. For example, : \begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A ''Solution (mathematics), solution'' to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. In the example above, a solution is given by the Tuple, ordered triple (x,y,z)=(1,-2,-2), since it makes all three equations valid. Linear systems are a fundamental part of linear algebra, a subject used in most modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A Nonlinear system, system of non-linear equations can often be Approximation, approximated by a linear system (see linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Rank (linear Algebra)
In linear algebra, the rank of a matrix is the dimension of the vector space generated (or spanned) by its columns. p. 48, § 1.16 This corresponds to the maximal number of linearly independent columns of . This, in turn, is identical to the dimension of the vector space spanned by its rows. Rank is thus a measure of the " nondegenerateness" of the system of linear equations and linear transformation encoded by . There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics. The rank is commonly denoted by or ; sometimes the parentheses are not written, as in .Alternative notation includes \rho (\Phi) from and . Main definitions In this section, we give some definitions of the rank of a matrix. Many definitions are possible; see Alternative definitions for several of these. The column rank of is the dimension of the column space of , while the row rank of is the dimension of the row space of . A fundamental resul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Coefficient Matrix
In linear algebra, a coefficient matrix is a matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations. Coefficient matrix In general, a system with linear equations and unknowns can be written as : \begin a_ x_1 + a_ x_2 + \cdots + a_ x_n &= b_1 \\ a_ x_1 + a_ x_2 + \cdots + a_ x_n &= b_2 \\ &\;\; \vdots \\ a_ x_1 + a_ x_2 + \cdots + a_ x_n &= b_m \end where x_1, x_2, \ldots, x_n are the unknowns and the numbers a_, a_, \ldots, a_ are the coefficients of the system. The coefficient matrix is the matrix with the coefficient as the th entry: : \begin a_ & a_ & \cdots & a_ \\ a_ & a_ &\cdots & a_ \\ \vdots & \vdots & \ddots & \vdots \\ a_ & a_ & \cdots & a_ \end Then the above set of equations can be expressed more succinctly as : A\mathbf = \mathbf where is the coefficient matrix and is the column vector of constant terms. Relation of its properties to properties of the equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Identity Matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or can be trivially determined by the context. I_1 = \begin 1 \end ,\ I_2 = \begin 1 & 0 \\ 0 & 1 \end ,\ I_3 = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end ,\ \dots ,\ I_n = \begin 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end. The term unit matrix has also been widely used, but the term ''identity matrix'' is now standard. The term ''unit matrix'' is ambiguous, because it is also used for a matrix of on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Coefficients
In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or any other type of expression. It may be a number without units, in which case it is known as a numerical factor. It may also be a constant with units of measurement, in which it is known as a constant multiplier. In general, coefficients may be any expression (including variables such as , and ). When the combination of variables and constants is not necessarily involved in a product, it may be called a ''parameter''. For example, the polynomial 2x^2-x+3 has coefficients 2, −1, and 3, and the powers of the variable x in the polynomial ax^2+bx+c have coefficient parameters a, b, and c. A , also known as constant term or simply constant, is a quantity either implicitly attached to the zeroth power of a variable or not attached to other variables in an expression; for example, the constant coefficients of the expressions above are the number 3 and the parameter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Marvin Marcus
Marvin David Marcus (July 31, 1927, Albuquerque, New Mexico – February 20, 2016, Santa Barbara, California) was an American mathematician, known as a leading expert on linear and multilinear algebra. Education and career From 1944 to 1946, Marvin Marcus served in the United States Navy. At the University of California, Berkeley, he graduated with a bachelor's degree in mathematics in 1950 and a Ph.D. in mathematics in 1953. His Ph.D. thesis ''The application of fixed-point theorems to the perturbation of ordinary differential equations'' was supervised by Stephen P. L. Diliberto. At the University of British Columbia (UBC), Marcus was an instructor from 1954 to 1955, an assistant professor from 1955 to 1956, and an associate professor from 1957 to 1962. At UBC he became a friend of Henryk Minc and supervised the master's thesis of Robert Charles Thompson. For the academic year 1956–1957 Marcus was on sabbatical in Washington, D.C. at the National Bureau of Standards, where h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |