What is a linear equation

What is a Linear Equation? A 5 Minute Introduction

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 existing 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 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.

As of linear operators, see also

https://www.physicsforums.com/insights/hilbert-spaces-relatives/
https://www.physicsforums.com/insights/tell-operations-operators-functionals-representations-apart/

 

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