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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 free/independent variable in a linear system

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Definition of basic/dependent/leading variable in a linear systemClose match
All echelon forms of a linear system have the same free variablesClose 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
Definition of echelon form of a linear systemClose match
A set of vectors is linearly independent if and only if the matrix of column vectors in reduced row-echelon form has every column as a pivot column.Close match
A homogeneous system has a nontrivial solution if and only if it has a free variable.Close match
TerminologyClose match
Linear systems and echelon matricesClose match
The nonzero rows of an echelon form of a matrix are linearly independent.Close match
A set of vectors is linearly independent if and only if the homogeneous linear system corresponding to the matrix of column vectors has only the trivial solution.Close match
The number of pivots in the reduced row echelon form of a consistent system determines whether there is one or infinitely many solutions.Close match
Basic terminologyClose match
Linear systems of equationsClose match
Parametric form of the solution set of a system of linear equationsClose match
Linear (in)dependenceClose match
A consistent system with more variables than equations has infinitely many solutions.Close match
Definition of inconsistent linear systemClose match
A linear system is equivalent to a vector equation.Close match
Definition of solution to a linear equationClose match
Gaussian elimination as a method to solve a linear systemClose match

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