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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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  • Geometric picture of a 2-by-2 linear system
  • Geometric picture of the solution set of a linear equation in 3 unknowns
  • Geometric picture of a 3-by-3 linear system
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  1. The geometry of linear systems

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