System of linear equations

A collection of linear equations involving shared variables. Its solutions are assignments of values that satisfy every equation at once.

Gaussian elimination: A method that uses row operations to solve linear systems or determine their structure. It systematically reduces the equations to expose their solutions or inconsistencies.
Linear equation: An equation in which variables appear only to the first power and are not multiplied together. Each equation contributes one linear constraint to the shared variables.
Kirchhoff's circuit laws: Rules stating that currents balance at circuit junctions and voltage changes sum to zero around loops. Circuit currents and voltages are often found by solving the resulting simultaneous equations.
Nonlinear system of equations: A collection of equations in shared variables that includes at least one nonlinear equation. Nonlinear terms remove the flat geometric structure that makes linear systems tractable.
Affine subspace: A translated vector subspace, consisting of a fixed point plus vectors in a subspace. Every nonempty solution set of a linear system is an affine subspace.
Augmented matrix: A matrix that records a linear system’s coefficients alongside its constant terms. It encodes the entire system in a form suited to elimination.
Matrix: A rectangular array of numbers or other entries arranged in rows and columns. The coefficient matrix organizes how each variable appears across the equations.
Leontief input-output model: An economic model that represents interindustry production requirements with a matrix equation. Its production levels are determined by a linear system linking sectors.
Inconsistent system: A system of equations with no assignment of variables satisfying all equations. It is the outcome when the system’s constraints contradict one another.
Condition number: A measure of how strongly a problem’s output can change when its input changes slightly. An ill-conditioned coefficient matrix makes computed solutions sensitive to small errors.

Linked from 17 pages