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Introduction to Matrices definitions

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  • Matrix
    A rectangular arrangement of numbers in rows and columns used to organize and represent information, especially from systems of equations.
  • Row
    A horizontal sequence of elements within a matrix, often corresponding to an equation in a system.
  • Column
    A vertical sequence of elements within a matrix, typically representing coefficients for a specific variable.
  • Augmented Matrix
    A matrix that includes both the coefficients and constants from a system of equations, separated by a vertical line to indicate equality.
  • Coefficient
    A numerical factor in front of a variable within an equation, represented as an entry in a matrix.
  • Constant
    A fixed value in an equation, usually appearing in the last column of an augmented matrix.
  • Row Operation
    A manipulation applied to the rows of a matrix, including swapping, scaling, or adding multiples of rows.
  • Row Echelon Form
    A matrix form with ones along the diagonal and zeros below, simplifying the process of solving systems by back substitution.
  • Reduced Row Echelon Form
    A matrix form with ones along the diagonal and zeros both above and below, allowing direct reading of solutions.
  • Diagonal
    The set of entries in a matrix running from the top left to the bottom right, often targeted for ones in special forms.
  • Gaussian Elimination
    A method using row operations to transform a matrix into row echelon form for solving systems of equations.
  • Gauss-Jordan Elimination
    A process that extends Gaussian elimination to achieve reduced row echelon form, yielding solutions without further work.
  • Back Substitution
    A technique for solving variables in a system by working upwards from the last equation after achieving row echelon form.
  • System of Equations
    A collection of two or more equations with shared variables, often represented and solved using matrices.
  • Notation
    A set of symbols and conventions used to represent matrix operations, such as subscripts for row identification.