LINEAR TRANSFORMATIONS, LINEAR MAPPINGS, LINEAR OPERATORS ARE LINEAR VECTOR FUNCTIONS ASSIGNING OBJECTS TO OBJECTS 

In courses in mathematical analysis one studies functions of one or more variables i.e. functions such as y = f(x), g = f(x,y), h = f(x,y,z), etc.. These functions can be regarded as functions of points or vectors in a space. For example, g = f(x,y) can be viewed as g = f ( ) where is a point or vector (x,y) in two dimensional space. Similarly h = f(x,y,z) can be viewed as h = f ( ) where is the point or vector (x,y,z) in three dimensional space. Thus we view them as functions of a vector argument. In the case of h = f ( ) where is the vector (x,y,z) the function assigns a number to the vector (x,y,z) in three dimensional space. This idea can be broadened to the idea of functions whose arguments may be, not just vectors in n-space, but other things – things like polynomials, continuous functions on some interval, infinite series, matrices, transformations, etc.. The idea can be broadened even further so that not only may the argument of the function be something other than a number or a vector in n-space but the value can also be something other than a number, also. Under the new, more abstract, definition a function has as its argument, “objects” and as its value, “objects” i.e. it assigns to the objects of one set objects of another set. Broadening the term “vector” to include objects of an arbitrary nature, we can define a vector function as a function that assigns a “vector” from one set to a “vector” of another set. We can then define a linear vector function (also known as a linear operator, a linear transformation and a linear mapping) as a function T: V W, whose domain V is one vector space and range W is another vector space, which maps vectors from V into W in a linear way, mapping ax + by into ax' + by' for all a and b if it maps vectors x and y into x' and y'. The “vectors” of V and W can be any objects of an arbitrary nature for which addition and multiplication by a number make sense.