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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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  • Examples of vector spaces#24095
  • Factorization of matrices#23622
  • F^n is a vector space.#24099
  • For invertible linear transformations A and B#24030
  • For matrices#23500
  • Formula for computing the least squares solution to a linear system#23949
  • Formula for computing the least squares solution to a linear system.#23948
  • Formula for diagonalizing a real 2-by-2 matrix with a complex eigenvalue.#23618
  • Formula for the coordinates of a vector with respect to an orthogonal/orthonormal basis.#23933
  • Formula for the coordinates of the projection of a vector onto a subspace#23944
  • Formula for the determinant of a 2-by-2 matrix.#23742
  • Formula for the determinant of a 3-by-3 matrix.#23743
  • Formula for the inverse of a 2-by-2 matrix.#23521
  • Formula for the least-squares linear fit to 2-dimensional data#23952
  • Formula for the (orthogonal) projection of one vector onto another vector#23939
  • Formula for the spectral decomposition for a symmetric matrix#23576
  • For n-by-n invertible matrices A and B#23523
  • Gaussian elimination as a method to solve a linear system#23421
  • Gauss-Jordan procedure to put a matrix into reduced row echelon form#23540
  • Geometric description of span of a set of vectors in R^n (or C^n)#23806
  • Geometric picture of a 2-by-2 linear system#23428
  • Geometric picture of a 3-by-3 linear system#23430
  • Geometric picture of the solution set of a linear equation in 3 unknowns#23429
  • Geometric properties of linear transformations#23851
  • Geometric properties of linear transformations on R^2#23853
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