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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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501–525 of 715 nodes
  • Matrix transpose commutes with matrix inverse.#23526
  • Matrix transpose is an involution.#23504
  • Matrix-vector multiplication is a linear transformation.#23856
  • Matrix-vector product is associative#23486
  • Matrix-vector products#23482
  • Multiplication#23487
  • Multiplication by a change of coordinates matrix converts representations for different bases.#23997
  • Multiplication by a Hermitian matrix commutes with the standard inner product on C^n.#23598
  • Multiplication of block/partitioned matrices#23569
  • Multiplicity#23695
  • Multiplying a row by a scalar multiplies the determinant by that scalar.#23739
  • Nilpotent matrices#23580
  • Non-example of a linear transformation#23850
  • Nonsingular matrices and equivalences#23640
  • Normal matrices#23599
  • Norm and length#23907
  • Notation for entry of matrix#23450
  • Notation for the set of m by n matrices#23453
  • Operations on matrices#23460
  • Orthogonality#23914
  • Orthogonality and projection#23893
  • Parametric form of the solution set of a system of linear equations#23419
  • Parametric vector form of the solution set of a system of linear equations#23420
  • Particular types of matrices#23535
  • Projection#23937
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Math#23168

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