Isomorphisms - Brief Description and Application
This topic is covered in section 4.8 of textbook for MA265. The basic idea of isomorphism has to do with structures that while appearing different are structurally the same.
The definition follows:
" Let V be a real vector space with operations + (addition) and *(scalar multiplication) and let W be a real vector space with operations + (addition) and *(scalar multiplication). A one-to-one function L mapping V onto W is called isomorphism (from the Greek isos meaning "the same" and morphos meaning structure) of V onto W if:
a) L(v+w)=L(v)+L(w) for v, w in V
b) L(c*v)=c*L(v) for v in V, c is a real number
(Elementary Linear Algebra with Applications, Kolman and Hill, 9th ed.)
NOTE: Mapping L onto W as satisfies the properties of a) and b) is a linear transformation covered in section 6.1 of the textbook for MA265
To put the definition of isomorphism into plain words we can say that an isomorphism is a linear transformation that is both one-to-one and onto.
We now must explain what one-to-one and onto functions are.
One-to-one function can be described as a function where for a distinct variable there exists a distinct value. Given function f(x)=a, for distinct value of x there exists a distinct value of a.
Onto function can be described as a function that uses all values of its domain. So if we are given R2 vector space which shall be a two-dimensional plane, then a function that exists in that plane shall exist for all coordinates of said 2-D plane.
Examples:
f(x) =x^2 - not onto and not one-to-one because there is no distinct f(x) for every distinct x. Such as f(x)=4 for x=2, x=-2. And since there is no negative value of f(x), f(x) is not onto.
f(x) = x^3 - onto and one-to-one because there is a distinct f(x) for every x. This function exists for the entire domain and is therefore onto. For a distinct x there is a distinct f(x) and is therefore one-to-one.
IMPORTANT - examples above are examples of one-to-one and onto functions. They are NOT examples of isomorphs.
Applications -
Systems analysis of artificial and natural systems. By looking for isomorphism in natural systems allows to understand intrinsic qualities and synergistic behavior of various biological systems. In Computer Science and Information Technology industry, similar systems analysis is performed. Comparison between relatively uninvolved and significantly less costly and their expensive counterparts becomes useful when one is able to isolate the isomorphisms of the intrinsic behavior of respective systems. At that point modeling real world systems in a simulated environment becomes very accurate and and a lot more budget friendly.
