Linear function

A function of the form $y=kx+b$. The main property of a linear function is: The increment of the function is proportional to the increment of the argument. Graphically a linear function is represented by a straight line.

A linear function in $n$ variables $x_1,\dots,x_n$ is a function of the form $$f(x) = a_1x_1+\cdots + a_nx_n +a,$$ where $a_1,\dots,a_n$ and $a$ are certain fixed numbers. The domain of definition of a linear function is the whole $n$-dimensional space of the variables $x_1,\dots,x_n$, real or complex. If $a=0$, the linear function is called a homogeneous, or linear, form.

If all variables $x_1,\dots,x_n$ and coefficients $a_1,\dots,a_n, a$ are real (complex) numbers, then the graph of the linear function in the $n+1$-dimensional (complex) space of the variables $x_1,\dots,x_n,y$ is the (complex) $n$-dimensional hyperplane $y = a_1x_1+\cdots + a_nx_n +a$, in particular, for $n=1$ it is a straight line in the plane (respectively, a complex plane in two-dimensional complex space).

The term "linear function" , or, more precisely, homogeneous linear function, is often used for a linear mapping of a vector space $X$ over a field $K$ into this field, that is, for a mapping $f:X\to K$ such that for any elements $x',x''\in X$ and any $\alpha',\alpha''\in K$, $$f(\alpha'x'+\alpha''x'') = \alpha'f(x') + \alpha''f(x''),$$ and in this case instead of the term "linear function" one also uses the terms linear functional and linear form.