# College Math Teaching

## October 7, 2016

### Now what is a linear transformation anyway?

Filed under: linear albegra, pedagogy — Tags: , — collegemathteaching @ 9:43 pm

Yes, I know, a linear transformation $L: V \rightarrow W$ is a function between vector spaces such that $L(V \oplus W) = L(V) \oplus L(W)$ and $L(a \odot V) = a \odot L(V)$ where the vector space operations of vector addition and scalar multiplication occur in their respective spaces.

Consider the set $R^+ = \{x| x > 0 \}$ endowed with the “vector addition” $x \oplus y = xy$ where $xy$ represents ordinary real number multiplication and “scalar multiplication $r \odot x = x^r$ where $r \in R$ and $x^r$ is ordinary exponentiation. It is clear that $\{R^+, R | \oplus, \odot \}$ is a vector space with $1$ being the vector “additive” identity and $0$ playing the role of the scalar zero and $1$ playing the multiplicative identity. Verifying the various vector space axioms is a fun, if trivial exercise.

Then $L(x) = ln(x)$ is a vector space isomophism between $R^+$ and $R$ (the usual addition and scalar multiplication) and of course, $L^{-1}(x) = exp(x)$.

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 $R$, 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 $L(x): R^+ \rightarrow R^+$ by $L(x) = x^r$, $r$ any real number.

Exercise 1: $L$ is a linear transformation.

Exercise 2: If we have ANY linear transformation $L: R^+ \rightarrow R^+$, let $L(e) = e^a$.
Then $L(x) = L(e^{ln(x)}) = L(e)^{ln(x)} = (e^a)^{ln(x)} = x^a$.

Exercise 3: we know that all linear transformations $L: R \rightarrow R$ are of the form $L(x) = ax$. These can be factored through:

$x \rightarrow e^x \rightarrow (e^x)^a = e^{ax} \rightarrow ln(e^{ax}) = ax$.

So this isn’t exactly anything profound, but it is fun! And perhaps it might be a way to introduce commutative diagrams.