Yes, I know, a linear transformation is a function between vector spaces such that and where the vector space operations of vector addition and scalar multiplication occur in their respective spaces.
Previously, I talked about this classical example:
Consider the set endowed with the “vector addition” where represents ordinary real number multiplication and “scalar multiplication where and is ordinary exponentiation. It is clear that is a vector space with being the vector “additive” identity and playing the role of the scalar zero and playing the multiplicative identity. Verifying the various vector space axioms is a fun, if trivial exercise.
Then is a vector space isomophism between and (the usual addition and scalar multiplication) and of course, .
Can we expand this concept any further?
Question: (I have no idea if this has been answered or not): given any, say, non-compact, connected subset of , is it possible to come up with vector space operations (vector addition, scalar multiplication) so as to make a given, say, real valued, continuous one to one function into a linear transformation?
The answer in some cases is “yes.”
Consider by , any real number.
Exercise 1: is a linear transformation.
Exercise 2: If we have ANY linear transformation , let .
Exercise 3: we know that all linear transformations are of the form . These can be factored through:
So this isn’t exactly anything profound, but it is fun! And perhaps it might be a way to introduce commutative diagrams.