What is a Linear Equation? A 5 Minute Introduction

[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 existing 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 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.
As of linear...

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[fresh_42]

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/