Rouché–Capelli theorem

The Rouché–Capelli theorem states that a linear system is consistent exactly when its coefficient and augmented matrices have equal rank. When consistent, its solution set has dimension equal to the number of unknowns minus that rank.

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Rank–nullity theorem: For a linear map between finite-dimensional vector spaces, rank plus nullity equals the dimension of its domain. It explains why the number of free variables is the number of unknowns minus the coefficient matrix’s rank.
Linear system: A collection of linear equations sharing the same unknowns. The theorem classifies consistency and solution dimension for this broad class of systems.
Consistency of a linear system: The property of a system of equations having at least one solution. The theorem provides a necessary and sufficient rank condition for consistency.
Eugène Rouché: A French mathematician whose work included a criterion for consistency of linear systems. His name is attached to the rank criterion in the theorem’s French designation.
Augmented matrix: A matrix formed by appending a linear system’s constants column to its coefficient matrix. Its rank reveals whether the constants impose a contradiction beyond the coefficient equations.
Matrix rank: The dimension of the column space of a matrix, equal to the dimension of its row space. Comparing the two ranks is the theorem’s consistency test.
Cramer’s rule: A formula expressing the unique solution of a square linear system using determinants, when its coefficient determinant is nonzero. It handles a narrow nonsingular case, while rank comparison also covers singular and rectangular systems.
Alfredo Capelli: An Italian mathematician known for work in algebra and the theory of linear equations. His name accompanies Rouché’s in the theorem’s common international name.
Gaussian elimination: A method for solving linear systems by applying row operations to transform their matrices. Row reduction makes both relevant ranks and any inconsistent row visible.
Linear independence: A set of vectors is linearly independent when only the trivial linear combination equals zero. Rank counts independent columns, which determine how many constraints the equations impose.