Gaussian elimination

Gaussian elimination is an algorithm that applies elementary row operations to a system of linear equations, reducing its coefficient matrix to row-echelon form so solutions can be found.

Elementary row operations: The three reversible operations on matrix rows: swapping rows, scaling a row by a nonzero number, and adding a multiple of one row to another. These operations preserve the solution set while simplifying the equations.
System of linear equations: A collection of linear equations that share a set of unknown variables. Elimination is designed to solve such systems through equivalent equation transformations.
Gauss–Jordan elimination: A row-reduction algorithm that continues past row-echelon form to reduced row-echelon form. Unlike Gaussian elimination, it eliminates entries above pivots as well as below them.
Linear programming: A method of optimizing a linear objective function subject to linear equality and inequality constraints. Its algorithms repeatedly solve linear systems involving constraints and candidate solution directions.
Reduced row-echelon form: A row-echelon matrix whose pivots equal one and are the only nonzero entries in their columns. Continuing row reduction to this form makes pivot and free variables explicit.
Row-echelon form: A matrix form in which nonzero rows precede zero rows and each leading entry lies to the right of the one above it. This is the triangular arrangement Gaussian elimination aims to produce.
Linear combination: An expression formed by multiplying vectors or other objects by scalars and adding the results. Row replacement combines equations using scalar multiples.
LU decomposition: A factorization that expresses a matrix as the product of a lower-triangular matrix and an upper-triangular matrix. It packages elimination steps for solving multiple systems with the same coefficient matrix.
Finite element method: A numerical method that approximates solutions to differential equations by assembling equations over small subregions. Discretization produces large linear systems that must be solved computationally.
Homogeneous system: A linear system whose constant terms are all zero. Elimination shows whether its only solution is zero or whether nonzero solutions exist.

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