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  • Linear systems of equations

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  • 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
  • The geometry of linear systems
  • Definition of consistent linear system
  • Definition of inconsistent linear system
  • Definition of homogeneous linear system of equations
  • 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.
  • The number of solutions to a linear system
  • Definition of basic/dependent/leading variable in a linear system
  • Definition of free/independent variable in a linear system
  • Theorem describing the vector form of sulutions to a linear system.
  • The solutions to a nonhomogeneous system are given by a particular solution plus the solutions to the homogeneous system.
  • Definition of ill-conditioned linear system
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  1. The solutions to a nonhomogeneous system are given by a particular solution plus the solutions to the homogeneous system.

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476–500 of 715 nodes
  • Linear systems and echelon matrices#23770
  • Linear systems and matrices#23751
  • Linear systems have 0#23439
  • Linear systems of equations#23411
  • Linear transformations#23839
  • LU decomposition#23624
  • Matrices#23447
  • Matrices act as a transformation by multiplying vectors#23855
  • Matrices as linear transformations#23854
  • Matrix addition is commutative and associative.#23463
  • Matrix adjoint is an involution.#23509
  • Matrix conjugation is an involution.#23466
  • Matrix describing a rotation of the plane#23887
  • Matrix diagonalization#23609
  • Matrix equations#23763
  • Matrix equivalence#23603
  • Matrix inverse is an involution.#23522
  • Matrix inverses are unique: if A and B are square matrices#23533
  • Matrix multiplication can be viewed as the dot product of a row vector of column vectors with a column vector of row vectors#23501
  • Matrix multiplication is associative.#23496
  • Matrix multiplication is distributive over matrix addition.#23497
  • Matrix multiplication is not commutative in general.#23499
  • Matrix representation of a composition of linear transformations is given by a matrix product#23862
  • Matrix-scalar multiplication is commutative#23472
  • Matrix-scalar product is commutative#23473
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Math#23168

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