An elastostatic boundary value problem for an isotropic-homogeneous media is a system of 15 independent equations and equal number of unknowns ( 3 equilibrium equations, 6 strain-displacement equations, and 6 constitutive equations ).
22.
There will be an infinitude of other solutions only when the system of equations has enough dependencies ( linearly dependent equations ) that the number of independent equations is at most " N " & minus; 1.
23.
But with " M " e " " N " the number of independent equations could be as high as " N ", in which case the trivial solution is the only one.
24.
The number of independent equations in a system equals the rank of the augmented matrix of the system & mdash; the system's coefficient matrix with one additional column appended, that column being the column vector of constants.
25.
In linear systems, indeterminacy occurs if and only if the number of independent equations ( the rank of the augmented matrix of the system ) is less than the number of unknowns and is the same as the rank of the coefficient matrix.
26.
Well, if it's linear equations, you'd get infinitely many solutions when the system has at least one solution and the number of variables is strictly larger than the number of linearly independent equations . talk ) 19 : 38, 2 December 2008 ( UTC)
27.
Where m _ { ij } is the i, j-th element of M . The sum is antisymmetric with respect to indices i, j, and since the left hand side is zero when i differs from j, this leaves n ( 2n-1 ) independent equations.
28.
The rank of a system of equations can never be higher than [ the number of variables ] + 1, which means that a system with any number of equations can always be reduced to a system that has a number of independent equations that is at most equal to [ the number of variables ] + 1.
29.
For example, in linear algebra if the number of constraints ( independent equations ) in a system of linear equations equals the number of unknowns then precisely one solution exists; if there are fewer independent equations than unknowns, an infinite number of solutions exist; and if the number of independent equations exceeds the number of unknowns, then no solutions exist.
30.
For example, in linear algebra if the number of constraints ( independent equations ) in a system of linear equations equals the number of unknowns then precisely one solution exists; if there are fewer independent equations than unknowns, an infinite number of solutions exist; and if the number of independent equations exceeds the number of unknowns, then no solutions exist.
