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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 basic/dependent/leading variable in a linear system

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Definition of free/independent variable in a linear systemClose match
All echelon forms of a linear system have the same free variablesClose match
Definition of echelon form of a linear systemClose match
TerminologyClose match
Basic terminologyClose match
Linear systems and echelon matricesClose match
The number of pivots in the reduced row echelon form of a consistent system determines the number of free variables in the solution set.Close match
Example of putting a matrix in echelon form and identifying the pivot columnsClose match
Definition of linear equationClose match
Definition of pivot positionClose match
Definition of coefficients of a linear equationClose match
Definition of pivotClose match
A homogeneous system has a nontrivial solution if and only if it has a free variable.Close match
A linear system is equivalent to a vector equation.Close match
Example of (echelon matrix/matrix in (row) echelon form)Close match
A linear system is equivalent to a matrix equation.Close match
Definition of coefficient matrix of a linear systemClose match
Definition of (echelon matrix/matrix in (row) echelon form)Close match
Linear systems and matricesClose match
Linear systems of equationsClose match

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