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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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The solutions to a nonhomogeneous system are given by a particular solution plus the solutions to the homogeneous system.

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The solutions of a homogeneous system are the pre-image (of 0) of a linear transformation.Close match
A homogeneous system has a nontrivial solution if and only if it has a free variable.Close match
Homogeneous linear systems are consistent.Close match
Definition of nontrivial solution to a homogeneous linear system of equationsClose match
Definition of homogeneous linear system of equationsClose match
The solutions to a homogeneous system of linear equations is a vector space.Close match
Definition of trivial solution to a homogeneous linear system of equationsClose match
A homogeneous system with more variables than equations has infinitely many solutions.Close match
Vector form of solutions to a consistent linear systemClose match
The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A.Close match
Equivalence theorem for nonsingular matrices: the equation Ax=b has a unique solution for all b.Close match
The solutions to a homogeneous linear differential equation is a vector space.Close match
Equivalence theorem for nonsingular matrices: the equation Ax=b has a solution for all b.Close match
Definition of solution to a linear equationClose match
Matrix equationsClose match
Example of solving a 3-by-3 homogeneous matrix equationClose match
Definition of solution vector of a linear systemClose match
A matrix equation is equivalent to a linear systemClose match
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