Linear function

A function of the form . 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 variables is a function of the form

where and are certain fixed numbers. The domain of definition of a linear function is the whole -dimensional space of the variables , real or complex. If , the linear function is called a homogeneous, or linear, form.

If all variables and coefficients are real (complex) numbers, then the graph of the linear function in the -dimensional (complex) space of the variables is the (complex) -dimensional hyperplane , in particular, for 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 over a field into this field, that is, for a mapping such that for any elements and any ,

and in this case instead of the term "linear function" one also uses the terms linear functional and linear form.