1. Linear Functions

Linear Function

A function $f$ with constant $\alpha$, where
$f(\alpha x)=\alpha f(x)$
is called a linear function.

Explanation

This is pretty simple if we regard the formula above as an equation of two linear functions $f_i$ and $g_i$, where
$f_1(x_1)+f_2(x_2)+...+f_n(x_n)+b_f=g_1(x_1)+g_2(x_2)+...+g_n(x_n)+b_g$
Over each term of $x$ is applied a certain (linear) transformation, usually a constant. We can expand this equation even further by equating $f_i(x_i)={\alpha }_i{x}_i$. and $g_i(x_i)={\beta }_i{x}_i$. We then have:${\alpha }_1{x}_1+{\alpha }_2{x}_2+...+{\alpha }_n{x}_n+b_f={\beta }_1{x}_1+{\beta }_2{x}_2+...+{\beta }_n{x}_n+b_g$This suggests that a linear equation (1) is equal to two linear combinations (in this case the LHS and RHS of the 2nd equation).