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.
Parents
Children
- Basic terminology
- Parametric form of the solution set of a system of linear equations
- Parametric vector form of the solution set of a system of linear equations
- Gaussian elimination as a method to solve a linear system
- Definition of echelon form of a linear system
- All echelon forms of a linear system have the same free variables
- Definition of equation operations on a linear system
- Equation operations on a linear system give an equivalent system.
- Definition of equivalent systems of linear equations
- The geometry of linear systems
- Definition of consistent linear system
- Definition of inconsistent linear system
- Definition of homogeneous linear system of equations
- Homogeneous linear systems are consistent.
- Definition of nontrivial solution to a homogeneous linear system of equations
- Definition of trivial solution to a homogeneous linear system of equations
- A homogeneous system has a nontrivial solution if and only if it has a free variable.
- The number of solutions to a linear system
- Definition of basic/dependent/leading variable in a linear system
- Definition of free/independent variable in a linear system
- Theorem describing the vector form of sulutions to a linear system.
- The solutions to a nonhomogeneous system are given by a particular solution plus the solutions to the homogeneous system.
- Definition of ill-conditioned linear system