Linear equation/Advanced

In mathematics, or more specifically algebra, a linear equation is an equation involving only polynomials of the first degree. Linear equations are ubiquitous in applications of mathematics. Equations involving a single variable appear in the simplest problems where one unknown quantity needs to be determined from other given information.

Quadratic equations occurring in applications typically involve real number coefficients. However, one can algebraically manipulate polynomial equations in the usual way as long as the coefficients can be added and multiplied together. Please see the main page for a discussion of linear equations with real coefficients.

The most general mathematical context that deals with systems of objects that can be added and multiplied together is ring theory. One can define polynomials, and in particular quadratic polynomial equations, as long as the coefficients are in a ring. The real numbers is an example of a ring. Another example, important in coding theory, is polynomials with coefficients in the ring $$\mathbb{Z}_2 = \{ \, \overline{0}, \overline{1} \, \}$$. You add and multiply in this ring in the same way you add or multiply the integers $$\{ \, 0, 1, \, \}$$ with one exception: since $$\mathbb{Z}_2$$ does not have a "two" in it, we set $$\overline{1} + \overline{1} = \overline{0}$$.

Solutions of linear equations
When working with polynomials over a specific ring $$R$$, one usually looks for solutions in the same ring $$R$$. The main exception to this is the most common case, where a polynomial has integer coefficients but one desires real number solutions. If, instead, one demands solutions of the same type as the polynomial coefficients, namely integers, the equation becomes a Diophantine equation. In this article, we assume that the desired solutions are in the same ring that the coefficients are drawn from.

Linear equations over a division ring
Every non-degenerate linear equation with real coefficients has a solution. This is no longer the case when the coefficients and solutions are in a general ring, but it is true if the ring is a division ring. The method of finding solutions looks formally the same as that when the coefficients are real numbers: after using algebraic operations to isolate one variable, every solution can be found by making an arbitrary choice of values for the remaining variables.