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  • Linear systems of equations

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  • Definition of consistent linear system
  • Definition of inconsistent linear system
  • 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.
  • Definition of basic/dependent/leading variable in a linear system
  • Definition of free/independent variable in a linear system
  • The solutions to a nonhomogeneous system are given by a particular solution plus the solutions to the homogeneous system.
  • 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
  • Definition of homogeneous linear system of equations
  • The number of solutions to a linear system
  • Vector form of solutions to a consistent linear system
  • Definition of ill-conditioned linear system
  • The geometry of linear systems

Children3
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  • A real or complex linear system has zero, one, or infinitely many solutions
  • A consistent system with more variables than equations has infinitely many solutions.
  • A homogeneous system with more variables than equations has infinitely many solutions.
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  1. The number of solutions to a linear system

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701–718 of 718 nodes
  • Transpose and adjoint#23502
  • Transpose commutes with scalar multiplication.#23511
  • Triangularization requires the characteristic polynomial to split#24076
  • Triangular matrices#23561
  • Two matrices of the same size are equivalent if and only if they have the same rank.#23607
  • Two vectors are dependent iff at least one is a scalar multiple of the other#23826
  • Unitary matrices#23587
  • Unitary matrices are invertible.#23589
  • Unitary matrices have orthogonal (orthonormal) rows/columns.#23590
  • Unitary matrices preserve inner products.#23591
  • Unitary matrices preserve orthogonal (orthonormal) bases.#23592
  • Using matrices to solve linear systems#23760
  • Vector form of solutions to a consistent linear system#23444
  • Vector space isomorphism is an equivalence relation.#24003
  • Vector spaces#23775
  • Vector sum/addition interpreted geometrically in R^n (or C^n)#23797
  • Vector sum/addition is commutative and associative#23787
  • Visualise a linear transformation on R^2 by looking at the image of a region#23852
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Container for Linear Algebra#24123
  • Math#23168
  • Test Flight Problem Set#23151

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