Search results
Results From The WOW.Com Content Network
An underdetermined linear system has either no solution or infinitely many solutions. For example, + + = + + = is an underdetermined system without any solution; any system of equations having no solution is said to be inconsistent. On the other hand, the system
By the Rouché–Capelli theorem, the system of equations is inconsistent, meaning it has no solutions, if the rank of the augmented matrix (the coefficient matrix augmented with an additional column consisting of the vector b) is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal ...
A solution of a linear system is an assignment of values to the variables ,, …, such that each of the equations is satisfied. The set of all possible solutions is called the solution set. [5] A linear system may behave in any one of three possible ways: The system has infinitely many solutions.
[1] [citation needed] An overdetermined system is almost always inconsistent (it has no solution) when constructed with random coefficients. However, an overdetermined system will have solutions in some cases, for example if some equation occurs several times in the system, or if some equations are linear combinations of the others.
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal ...
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the ...
The state-transition matrix is used to find the solution to a general state-space representation of a linear system in the following form ˙ = () + (), =, where () are the states of the system, () is the input signal, () and () are matrix functions, and is the initial condition at .