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Linear systems of equations
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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
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A homogeneous system has a nontrivial solution if and only if it has a free variable.
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