Linearity

A linear combination of $x_{1}, x_{2}, \dots, x_{n}$ is an expression

a_{1} x_{1} + a_{2} x_{2} + \dots + a_{n} x_{n}

where $x_{i}$ 's are indeterminates or variables; and $a_{i}$ 's are coefficients and belong to a field $𝕂$ .

In the above definition and in the remainder of these notes the field $𝕂$ will mostly be either the field of real numbers $ℝ$ or the field of complex numbers $ℂ$ , but in general any field will do, except in special cases when $ℂ$ (or a complete field) is required. We will strive to make a note whenever $ℂ$ is required.

\begin{matrix} linear & non-linear \\ x + y + z & tan (x) + x + (-1) z \\ 2 ⅈ x + x + (-1) z & x + x y + (-1) z \\ x + (1 - 7 ⅈ) y + \sqrt{-1} z & x^{2} + sin (y) x + (-1) z \\ (\int_{0}^{4} x ⅆ x) x_{1} + (1 - 7 ⅈ) x_{2} + \sqrt{-1} x_{3} & sin (x) + sin (y) + sin (z) \end{matrix}

Often $0 x$ is omitted. Likewise instead of $1 x$ one writes just $x$ and instead of $(-1) x$ one writes $- x$ .

It is important to keep in mind that there is a difference between linear in $x$ and linear in $sin x$ . While both are possible and have usefulness here we focus on linearity and unless otherwise specified will restrict our attention to linear in a set of variables $𝒳 = {x_{1}, x_{2}, \dots, x_{n}}$