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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 (X) where X is a
point or vector (x,y) in two dimensional space. Similarly h = f(x,y,z) can be viewed as h = f (X)
where X 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 (X) where X 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.

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