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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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Definition of echelon form of a linear system

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Linear systems and echelon matricesClose match
Echelon matricesClose match
Definition of (row) echelon form of a matrixClose match
All echelon forms of a linear system have the same free variablesClose match
Definition of (echelon matrix/matrix in (row) echelon form)Close match
Example of putting a matrix in echelon formClose match
Definition of matrix in reduced row echelon formClose match
Definition of reduced row echelon form of a matrixClose match
Every matrix is row-equivalent to a matrix in reduced row echelon form.Close match
Example of putting a matrix in echelon form and identifying the pivot columnsClose match
Example of (echelon matrix/matrix in (row) echelon form)Close match
Gauss-Jordan procedure to put a matrix into reduced row echelon formClose match
The nonzero rows of an echelon form of a matrix are linearly independent.Close match
Definition of free/independent variable in a linear systemClose match
Definition of basic/dependent/leading variable in a linear systemClose match
The echelon form can be used to determine if a linear system is consistent.Close match
Example of using the echelon form to determine if a linear system is consistent.Close match
Every matrix is row-equivalent to only one matrix in reduced row echelon form.Close match
Gaussian elimination as a method to solve a linear systemClose match
TerminologyClose match

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