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Linear Systems
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
To find a way to get a (near) LU decomposition of matrices that require pivoting–which, because of stability concerns, is the typical case, unlike the specially prepared examples in the previous section–let's revisit Gaussian elimination. We perform Gaussian elimination using two of the three elementary row operations. Each elementary row operation corresponds to an elementary matrix that implements that operation by premultiplication; this matrix is always the matrix that arises by applying the elementary row operation to the identity matrix. That is, if A is an n×n matrix then performing Ri↔Rj to A is equivalent to forming the matrix product PA where P is the matrix obtained from In by interchanging rows i and j of it, and similarly for the other row operations.
Matrix Decomposition with Applications
Published in Crista Arangala, Exploring Linear Algebra, 2019
Recall from Lab 2, that we can perform elementary row operations on a matrix by multiplying the matrix by elementary matrices. The steps of the LU Decomposition of matrix A are to Determine elementary matrices to transform A into an upper triangular matrix, U. That is find E1,E2, …, Ek such that Ek … E2E1A = U.Write A = (Ek … E2E1)−1U = LU where L = (Ek … E2E1)−1.
Vectors, Matrices, and Systems of Linear Equations
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
The matrix A is row equivalent to the matrix B if there is a sequence of elementary row operations that transforms A into B. The reduced row echelon form of A, RREF(A), is the matrix in reduced row echelon form that is row equivalent to A. Arow echelon form ofA is any matrix in row echelon form that is row equivalent to A. The rank of A, denoted rank A or rank(A), is the number of leading entries in RREF(A). If A is in row echelon form, the positions of the leading entries in its nonzero rows are called pivot positions and the entries in those positions are called pivots. A column (row) that contains a pivot position is a pivot column (pivot row).
A case study of in-service teachers’ errors and misconceptions in linear combinations
Published in International Journal of Mathematical Education in Science and Technology, 2022
Lillias Hamufari Natsai Mutambara, Sarah Bansilal
All the other teachers in this category came up with the appropriate augmented matrix on which they carried out row reduction as in Step 4a of Table 1, but were not able to bring it to row echelon form. An analysis of the augmented matrix shows that only three elementary row operations were required to get the matrix to reduced row echelon form, yet the teachers struggled with the process. There were 20 teachers who made calculation errors or applied inappropriate row operations in working with the correct augmented matrix. Eight teachers made errors because they applied incorrect or inappropriate row operations, while 12 teachers made careless errors in manipulating the numbers. Some of these careless errors led to the teachers obtaining unique solutions, and without checking, they incorrectly deduced that the original vector v could be expressed as a linear combination of the three vectors and
A study of calculus students’ difficulties, approaches and ability to solve multivariable optimization problems
Published in International Journal of Mathematical Education in Science and Technology, 2022
A common theme that emerges from studies that have investigated students’ reasoning about systems of equations in linear algebra at the undergraduate level is that students tend to lack a conceptual understanding of the methods used to solve these systems of equations (cf., De Vries & Arnon, 2004; Kazunga & Bansilal, 2020; Maharaj, 2018; Ndlovu & Brijlall, 2019; Trigueros et al., 2007). Ndlovu and Brijlall (2019) reported on pre-service teachers who exhibited procedural fluency when using Cramer’s rule to solve a system of two equations in two unknowns in one task, but could not explain why this rule could not be used to solve a system of four equations in three unknowns in another task. These researchers argued that findings from their study shows that the ‘algorithm [Cramer’s rule] was instrumentally understood’ (p. 6) by the pre-service teachers which is why ‘they struggled to conceptualize the application of the algorithm beyond carrying out procedures’ (p. 6). Similar student difficulties were observed in Maharaj’s (2018) study that examined students’ ability to solve a system of three equations in three unknowns by first writing the augmented matrix for the system and then applying elementary row operations. In addition to reporting on students’ fluency with algebraic procedures for solving systems of equations, findings by De Vries and Arnon (2004) and Trigueros et al. (2007) indicate that students occasionally confuse solutions to systems of equations with the constant terms in the systems.
The feedback invariant measures of distance to uncontrollability and unobservability
Published in International Journal of Control, 2022
Nicos Karcanias, Olga Limantseva, George Halikias
Given the set of polynomials of (29) with a generalised resultant , the following properties hold true:The necessary and sufficient condition for a set of polynomials to be coprime is that Let be the GCD of P. Then If we reduce , by using elementary row operations, to its row echelon form, the last non-vanishing row defines the coefficients of the GCD.