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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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  • Abstract vector spaces#23953
  • A consistent system with more variables than equations has infinitely many solutions.#23440
  • Adding a multiple of one row to another row does not change the determinant.#23737
  • Addition#23461
  • A determinant is a multilinear function#23736
  • A diagonalizable matrix is diagonalized by a matrix having the eigenvectors as columns.#23615
  • Adjoining an element not in the span of a linearly independent set gives another linearly independent set.#23983
  • A homogeneous system has a nontrivial solution if and only if it has a free variable.#23437
  • A homogeneous system with more variables than equations has infinitely many solutions.#23441
  • Algebraic properties of R^n (or C^n)#23777
  • Algorithm for computing an LU decomposition#23626
  • A linear system is equivalent to a matrix equation.#23753
  • A linear system is equivalent to a vector equation.#23752
  • A linear transformation has a representation as an upper triangular matrix.#24076
  • A linear transformation has the same eigenvalues and eigenvectors as any matrix representation.#24075
  • A linear transformation is determined by its action on a basis.#23866
  • A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.#24054
  • A linear transformation is given by a matrix whose columns are the images of the standard basis vectors#23858
  • A linear transformation is given by a matrix with respect to a given basis.#23859
  • A linear transformation is given by multiplying by its matrix representation with respect to bases of the spaces#24069
  • A linear transformation is injective on its generalized range space.#24040
  • A linear transformation is invertible if and only if it is injective and surjective#24067
  • A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.#24066
  • A linear transformation is surjective if and only if the columns of its matrix span the codomain.#23885
  • A linear transformation is surjective if and only if the image of a basis is a spanning set#24038
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