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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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  • An n-by-n matrix is diagonalizable if and only if the union of the basis vectors for the eigenspaces is a basis for R^n (or C^n).#23616
  • An n-by-n matrix nas n (complex) eigenvalues#23698
  • An n-by-n matrix with n distinct eigenvalues is diagonalizable.#23617
  • A nonempty subset of a vector space is a subspace if and only if it is closed under linear combinations#23965
  • A nonsingular matrix can be written as a product of elementary matrices.#23558
  • An orthogonal set of nonzero vectors is linearly independent.#23927
  • Any linearly independent set can be expanded to a basis for the (sub)space#23984
  • Any matrix times the 0 matrix equals the 0 matrix.#23491
  • Any vector space is the direct sum of the generalized kernel and gneralized range of a linear transformation on that space.#24061
  • Application Leontief input-output analysis#24118
  • Applications#24113
  • Applications of band matrices#24116
  • Applications to cubic spline#24120
  • Applications to differential equations#24114
  • Applications to error-correcting code#24117
  • Applications to Markov chains#24115
  • Applications to voting and social choice#24119
  • A scalar multiple of a linear transformation is a linear transformation#24018
  • A set is a basis if each vector can be written uniquely as a linear combination.#23988
  • A set is linearly independent if and only if the set of coordinate vectors with respect to any basis is linearly independent.#23981
  • A set of nonzero vectors contains (as a subset) a basis for its span.#23985
  • A set of two vectors is linearly dependent if and only if neither is a scalar multiple of the other.#23826
  • A set of vectors containing fewer elements than the dimension of the space cannot span#23979
  • A set of vectors containing more elements than the dimension of the space must be linearly dependent#24012
  • A set of vectors is linearly independent if and only if the homogeneous linear system corresponding to the matrix of column vectors has only the trivial solution.#23823
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