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Matrices
Published in Nita H. Shah, Foram A. Thakkar, Matrix and Determinant, 2020
Nita H. Shah, Foram A. Thakkar
Hence, when the constants are included, the resulting matrix is referred to as an ‘augmented matrix’. Augmented matrices can be used to find solutions to linear equations. Also, one can use shorthand to describe matrix operations as below: R1 represents row 1; similarly R2 represents row 2, …Add row 1 to row 3 and replace row 3 with that sum as R1+R3→R3R1↔R2 means interchanging row 1 and row 2.Multiplication of row 2 by 12 is written as 12R2→R2.
Systems of Linear Equations
Published in James R. Kirkwood, Bessie H. Kirkwood, Elementary Linear Algebra, 2017
James R. Kirkwood, Bessie H. Kirkwood
A system of linear equations has no solution if and only if one of the rows in the row reduced matrix of the augmented matrix for the system is 00⋯01.
Systems of Linear Equations
Published in James R. Kirkwood, Bessie H. Kirkwood, Linear Algebra, 2020
James R. Kirkwood, Bessie H. Kirkwood
A system of linear equations has no solution if and only if one of the rows in the row reduced matrix of the augmented matrix for system is(00⋯01).
Visualizing Laguerre polynomials as a complete orthonormal set for the inner product space ℙ n
Published in International Journal of Mathematical Education in Science and Technology, 2023
In the second step, multiple approaches could be embraced for establishing the basisness. One way is by entering the MATLAB syntax L0 = [1;0;0;0;0;0]; L1 = [1;−1;0;0;0;0]; L2 = [1;−2;0.5;0;0;0]; L3 = [1;−3;1.5;−1/6;0;0]; L4 = [1;−4;3;−2/3;1/24;0]; L5 =[1;−4;3;−2/3;1/24;0]; w = [0;0;0;0;0;0]; A = [L0 L1 L2 L3 L4 L5 w]; B = rref(A) to obtain the augmented matrix A = [L0 L1 L2 L3 L4 L5 w] and the reduced-row echelon form of the augmented matrix via the syntax B = rref(A), respectively (Figure 10(a)). We note that the trivial solution is the only solution therefore we obtain a linearly independent set due to the emerging identity matrix as the rref of the coefficient matrix []. Another similar approach could be adopted by simply introducing the coefficient matrix instead of the augmented matrix (Figure 10(b)).
Optimization of the right-hand sides of nonlocal conditions of a controllable system with multipoint and integral objective functional
Published in Optimization, 2022
K. R. Aida-zade, V. M. Abdullayev
Therefore, it is possible to extract from the augmented matrix α the matrix (minor) of rank formed by columns of the matrix α: Suppose is the number of the column of the matrix , , included as the ith column in the matrix , i.e. the ith column of the matrix is the th column of the matrix .
Multi-copy dynamic cloud data auditing model based on IMB tree
Published in Enterprise Information Systems, 2021
Yifan Zhang, Guoyuan Lin, Hao Gu, Fu Zhuang, Guoying Wei
In the system of Equations (8), if the rank of the coefficient matrix formed by the entire system of Equations is M and equal to the rank of its augmented matrix, then a unique solution can be obtained by linear algebra method, which means the physical content of the challenge data block is leaked to TTP. In MCAM, the evidence coefficient of each data block is determined by TTP and sent to the cloud, which prevents the cloud from impersonating aggregate evidence by generating a fixed random number sequence. In addition, contains the aggregation of both physical and logical evidence. After M times of repeated audit set, the Equations (9) can be obtained.