Definition:Linear Equation

Definition

A linear equation is an equation in the form:

$b = a_1 x_1 + a_2 x_2 + \cdots + a_n x_n$

where all of $a_1, \ldots, a_n, x_1, \ldots x_n, b$ are elements of a given field.

The point is that all the indices of the $x$ and $y$ terms in such an equation are $1$.

Examples

Arbitrary Example

The equation:

$x + 3 y + 2 z = 7$

is a linear equation in $3$ variables.

Arbitrary Non-Linear Equation

The equation:

$b = a_1 {x_1}^2 + a_2 {x_2}^2 + a_3 {x_3}^2$

is not a linear equation, because all the $x$ terms are squared.

Also see

• Results about linear equations can be found here.

Sources

• 1954: A.C. Aitken: Determinants and Matrices (8th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions and Fundamental Operations of Matrices: $2$. Linear Equations and Transformations
• 1987: A.G. Hamilton: A First Course in Linear Algebra ... (next): $1$: Gaussian Elimination
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): linear
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): linear
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): linear equation
• 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $4$: Lure of the Unknown: Equations
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): linear equation