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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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  • 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.#23824
  • A subset of a linearly independent set is linearly independent.#23980
  • A vector can be written uniquely as a linear combination of vectors from independent subspaces.#23973
  • A vector can be written uniquely as a sum of a vector in a subspace and a vector orthogonal to the subspace.#23934
  • A vector is in the orthogonal complement of a subspace if and only if it is orthogonal to every vector in a basis of the subspace.#23923
  • Axioms of a vector space#23798
  • Bases#23828
  • Basic properties#23957
  • Basic properties of linear transformations#23863
  • Basic terminology#23412
  • Basic terminology and notation#23448
  • Block matrices#23567
  • Canonical forms of matrices#23608
  • Change of coordinates matrices are invertible#23998
  • Characteristic and minimal polynomials#23699
  • C^n is a vector space.#24098
  • Cofactors#23716
  • Composition#24022
  • Conjugating by a change of coordinates matrix converts matrix representations with respect to different bases.#23999
  • Conjugation#23464
  • Container for Linear Algebra#24123
  • Coordinates#23989
  • Coordinate vector spaces#23776
  • Cramer's rule#23722
  • Definition and terminology#23954
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