Linear systems of equations

Linear systems of equations are a foundational topic in linear algebra. A linear system consists of multiple linear equations involving the same set of variables. Solving such a system means finding all values of the variables that satisfy every equation simultaneously.

Key concepts

• Linear equations — equations of the form $a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b$
• Solution set — the collection of all tuples that satisfy every equation
• Gaussian elimination — the primary algorithm for solving linear systems by transforming to echelon form
• Consistency — whether a system has at least one solution (consistent) or none (inconsistent)
• Homogeneous systems — systems where all constant terms are zero ($Ax = 0$)
• Geometry — each linear equation represents a hyperplane; solutions are intersections of these hyperplanes

Every linear system falls into exactly one of three categories: no solution, exactly one solution, or infinitely many solutions.