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Finite Element Concepts In One-Dimensional Space
Published in Steven M. Lepi, Practical Guide to Finite Elements, 2020
In certain cases the system of equations cannot be solved, since some systems have no unique solution. This will be illustrated in the next example in which Gaussian elimination is applied to the finite element equilibrium equations from Example 2.2, which were also used in the discussion of simultaneous equations in this section, in the form of (2.7.2):
Algebraic Structures II (Vector Spaces and Finite Fields)
Published in R. Balakrishnan, Sriraman Sridharan, Discrete Mathematics, 2019
R. Balakrishnan, Sriraman Sridharan
It is possible that such a system of equations has no solution at all. For example, consider the system of equations X1−X2+X3=2X1+X2−X3=03X1=6. From the last equation, we get X1 = 2. This, when substituted in the first two equations, yields −X2 + X3 = 0, X2 − X3 = −2 which are mutually contradictory. Such equations are called inconsistent equations.
Linear Algebra
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Fatemeh Hamidi Sepehr, Erchin Serpedin
Often it is necessary to extend the concept of matrix invertibility to a wider class of rank-deficient matrices by means of concepts such as generalized inverse or pseudoinverse of a matrix. One of the major applications of the concept of generalized inverse is in solving systems of equations of the form: Ax = b, where matrix A is neither square nor nonsingular. Such a system of equations may not admit any solution or may admit an infinite number of solutions.
Dynamic Multivariate Functional Data Modeling via Sparse Subspace Learning
Published in Technometrics, 2021
Chen Zhang, Hao Yan, Seungho Lee, Jianjun Shi
With this assumption, can be recovered as a sparse solution of the multilinear regression equation , with the regression coefficients that have only for , and bjr = 0 otherwise. Notably, for a system with equations such as (3), may have infinite number of solutions. Similar to the sparse subspace learning, we can obtain the optimal solution by minimizing the objective function with the lq-norm of the solution, that is,
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
T25 recognized that an inconsistent system of equations implied that the set of vectors were not a linear combination of the given vector one, however ,in her written response she stopped at that point of getting the inconsistent equation. Her comments also reveal another misconception that the only time the linear combination equation in Step 1 of Table 1 could be solved was if the solution to the system of equations generated by the linear combination relationship, was unique. However, this is not necessarily true because if the system of equations has an infinite number of solutions then there can be an infinite number of ways of expressing the given vector as a linear combination of the set of vectors.
Saving runs in fractional factorial designs
Published in Quality Engineering, 2019
Pere Grima, Lourdes Rodero, Xavier Tort-Martorell
Estimating two missing responses must be done through a system of two equations, each one based on an interaction. However, not all the systems that can appear are consistent. For example, from the interactions of and equaled to zero, we obtain an inconsistent system of equations. However, with and , we have: