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Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set. For three variables, each linear equation determines a plane in three-dimensional space, and the solution set is the intersection of these planes. Thus the ...
Systems of linear equations form a fundamental part of linear algebra. Historically, linear algebra and matrix theory have been developed for solving such systems. In the modern presentation of linear algebra through vector spaces and matrices, many problems may be interpreted in terms of linear systems.
Linear and non-linear equations. In the case of a single equation, the "solver" is more appropriately called a root-finding algorithm. Systems of linear equations. Nonlinear systems. Systems of polynomial equations, which are a special case of non linear systems, better solved by specific solvers. Linear and non-linear optimisation problems
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 ...
Plot of a quadratic equation (red) and a linear equation (blue) that do not intersect, and consequently for which there is no common solution. In the above example, a solution exists. However, there are also systems of equations which do not have any solution. Such a system is called inconsistent. An obvious example is
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. For if there are at least as many independent equations as unknowns, that will eliminate any stretches of overlap ...
It is inconsistent if and only if 0 = 1 is a linear combination (with polynomial coefficients) of the equations (this is Hilbert's Nullstellensatz). If an underdetermined system of t equations in n variables (t < n) has solutions, then the set of all complex solutions is an algebraic set of dimension at least n - t. If the underdetermined ...
Conversely, every line is the set of all solutions of a linear equation. The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line. If b ≠ 0, the line is the graph of the function of x that has been defined in the preceding ...