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Linear systems of equations

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Definition of consistent linear system
Definition of inconsistent linear system
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.
Definition of basic/dependent/leading variable in a linear system
Definition of free/independent variable in a linear system
The solutions to a nonhomogeneous system are given by a particular solution plus the solutions to the homogeneous system.
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
Definition of homogeneous linear system of equations
The number of solutions to a linear system
Vector form of solutions to a consistent linear system
Definition of ill-conditioned linear system
The geometry of linear systems

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Gaussian elimination as a method to solve a linear system

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