# What is a linear equation

1. Jul 23, 2014

### Greg Bernhardt

Definition/Summary

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

Equations

$$Mx+B=0$$

Extended explanation

Since $M\neq 0$ the solution is given by
$$x=-B/M\;.$$

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
$$\det(M)\neq 0\;,$$
and the solution is given by
$$\vec x = -M^{-1}\vec B\;,$$
where $M^{-1}$ 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
$$M=\left(\frac{i}{\hbar}\frac{\partial}{\partial t}-\hat H\right)\;,$$
where $\hat H$ is the hamiltonian.

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