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What is a linear equation

  1. Jul 23, 2014 #1
    Definition/Summary

    A first order polynomial equation in one variable, its general form is [itex]Mx+B=0[/itex] where x is the variable. The quantities M, and B are constants and [itex]M\neq 0[/itex].

    Equations

    [tex]
    Mx+B=0
    [/tex]

    Extended explanation

    Since [itex]M\neq 0[/itex] the solution is given by
    [tex]
    x=-B/M\;.
    [/tex]

    The variable x does not have to be a number. For example, x and B could be vectors and M could be a matrix.

    In this case the condition for a solution to exist is
    [tex]
    \det(M)\neq 0\;,
    [/tex]
    and the solution is given by
    [tex]
    \vec x = -M^{-1}\vec B\;,
    [/tex]
    where [itex]M^{-1}[/itex] is the matrix inverse of M.

    Another (more abstract) example, is the Green's function equation for the time-dependent Schrodinger equation. In this case x is a Green's function, and B is a (dirac) delta function in time, and M is the operator
    [tex]
    M=\left(\frac{i}{\hbar}\frac{\partial}{\partial t}-\hat H\right)\;,
    [/tex]
    where [itex]\hat H[/itex] is the hamiltonian.

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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