This is nothing new; it is an example for undergraduates.

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.

Now consider the function with domain . (here: is the natural logarithm function). Now and . This shows that (the range has the usual vector space structure) is a linear transformation.

What is even better: which shows that so is one to one (of course, we know that from calculus).

And, given so is also onto (we knew that from calculus or precalculus).

So, is isomorphic to with the usual vector operations, and of course the inverse linear transformation is .

Upshot: when one asks “is F a linear transformation or not”, one needs information about not only the domain set but also the vector space operations.

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